understanding and manipulating electronic quantum coherence in photosynthetic light-harvesting

Understanding and manipulating electronic quantum coherence inphotosynthetic light-harvesting

Copyright 2013by

Stephan Owen Steele Hoyer

Understanding and manipulating electronic quantum coherence inphotosynthetic light-harvesting

by


A dissertation submitted in partial satisfaction of therequirements for the degree of

Doctor of Philosophy

in

Physics

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor K. Birgitta Whaley, Co-chairProfessor Joel E. Moore, Co-chair

Professor Hartmut HäffnerProfessor Naomi Ginsberg

Fall 2013

1

Abstract

Understanding and manipulating electronic quantum coherence in photosyntheticlight-harvesting

by


Doctor of Philosophy in Physics

University of California, Berkeley

Professor K. Birgitta Whaley, Co-chair

Professor Joel E. Moore, Co-chair

Ultrafast spectroscopy experiments show that photosynthetic systems can preserve quantumbeats in the process of electronic energy transfer between pigments, even at room tem-perature. But what does this discovery imply for biology – and for quantum mechanics?This dissertation examines photosynthesis through the tools of quantum information sci-ence. We evaluate to what extent photosynthesis can be thought of as a type of quantumtechnology, and consider how we can apply experimental tools widely used for quantum tech-nologies (state tomography and coherent control) to photosynthesis. Throughout, we use theFenna-Matthews-Olson (FMO) complex of greens sulfur bacteria as a model photosyntheticpigment-protein complex.

In Part I, we evaluate two mechanisms for the possible biological relevance of quantum co-herent motion. First, we consider the extent to which dynamics in light-harvesting systemsexhibit the quantum speedups characteristic of quantum algorithms and quantum walks.Second, we demonstrate a ratchet effect in electronic energy transfer enabled by partiallycoherent dynamics. To do so, we build a new model of energy transfer between weakly cou-pled light-harvesting complexes to understand how electronic coherence arises under naturalconditions.

In Part II, we present proposals for experimental probes of light harvesting dynamicswith ultrafast spectroscopy. We show how the signal from pump-probe spectroscopy can beformally inverted to determine the excited state electronic density matrix. Finally, we exam-ine the feasiblity of coherent control experiments on light-harvesting systems, and providetwo realistic targets feasible with present-day technology.

i

Contents

Note on previously published work iv

1 Introduction to excitonic energy transfer and non-linear spectroscopy 11.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Energy transfer dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Non-linear optical spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Flexible simulation of dynamics and spectroscopy . . . . . . . . . . . . . . . 7

I Biological relevance of quantum coherence 9

2 Limits of quantum speedup in photosynthetic light harvesting 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Fenna-Matthews-Olson complex . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Coherent dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Limits of quantum speedup . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6 Diffusive transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.8 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.8.1 FMO Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.8.2 Reduced hierarchy equations model for FMO . . . . . . . . . . . . . . 232.8.3 Decay of coherences in linear transport . . . . . . . . . . . . . . . . . 24

3 Spatial propagation of excitonic coherence enables ratcheted energy trans-fer 263.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Spatial propagation of coherence . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Coherent versus thermal transport in a model dimer with an energy gradient 333.4 Design for biomimetic ratchet . . . . . . . . . . . . . . . . . . . . . . . . . . 343.5 Role of coherent energy transport in the Fenna-Matthews-Olson complex . . 383.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

ii

3.7 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.7.1 Extending multichromophoric Förster theory . . . . . . . . . . . . . . 403.7.2 Weak system-bath coupling . . . . . . . . . . . . . . . . . . . . . . . 423.7.3 Proof that classical Markovian transport is unbiased . . . . . . . . . . 443.7.4 Ratchet methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.7.5 FMO Hamiltonian and singular value decompositions . . . . . . . . . 50

II Probing and controlling quantum coherence 54

4 Inverting pump-probe spectroscopy for state tomography of excitonicsystems 554.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Recipe for pump-probe spectroscopy . . . . . . . . . . . . . . . . . . . . . . 57

4.2.1 Detection scheme and probe convolution . . . . . . . . . . . . . . . . 584.2.2 Pump-probe response function . . . . . . . . . . . . . . . . . . . . . . 60

4.3 Inversion protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3.1 Deconvolution of the pump-probe signal . . . . . . . . . . . . . . . . 614.3.2 Obtaining the quantum state . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Example: dimer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.4.1 Analytical calculation of pump-probe response . . . . . . . . . . . . . 644.4.2 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4.3 Response function inversion . . . . . . . . . . . . . . . . . . . . . . . 674.4.4 State tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.5 Scaling to larger systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.7 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.7.1 Derivation of pump-probe signal . . . . . . . . . . . . . . . . . . . . . 784.7.2 Positivity of ρ(2)

e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.7.3 Tikhonov regularization . . . . . . . . . . . . . . . . . . . . . . . . . 784.7.4 Parameter selection for Tikhonov regularization . . . . . . . . . . . . 804.7.5 Alternative formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 814.7.6 Triple deconvolution to obtain the full 3rd-order response function . . 814.7.7 Mathematica code for derivation in Sec. 4.4.1 . . . . . . . . . . . . . 84

5 Realistic and verifiable coherent control of excitonic states in a lightharvesting complex 855.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Model for laser excitation of FMO . . . . . . . . . . . . . . . . . . . . . . . . 875.3 Optimal control methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.4 Limits of coherent control for initial state preparation in FMO . . . . . . . . 925.5 Authentication of coherent control in pump-probe spectra . . . . . . . . . . . 98

iii

5.6 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.7 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.7.1 Orientational average . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.7.2 Optimal equation-of-motion phase-matching approach for third-order

ultrafast spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 104

References 108

iv

Note on previously published work

Chapters 2-5 in this dissertation are adaptations of the following published papers:

• Stephan Hoyer, Mohan Sarovar and K. Birgitta Whaley, “Limits of quantum speedupin photosynthetic light harvesting,” New J. Phys. 12, 065041 (2010).

• Stephan Hoyer, Akihito Ishizaki and K. Birgitta Whaley, “Spatial propagation of exci-tonic coherence enables ratcheted energy transfer,” Phys. Rev. E 86, 041911 (2012).

• Stephan Hoyer and K. Birgitta Whaley, “Inverting pump-probe spectroscopy for statetomography of excitonic systems,” J. Chem. Phys. 138, 164102 (2013).

• Stephan Hoyer, Filippo Caruso, Simone Montangero, Mohan Sarovar, Tommaso Calarco,Martin B. Plenio and K. Birgitta Whaley, “Realistic and verifiable coherent con-trol of excitonic states in a light harvesting complex,” submitted to New J. Phys.,arXiv:1307.4807.

Sections 4.7.6 and 5.7.2 present prosposals that were not part of these published works.

1

Chapter 1

Introduction to excitonic energy transferand non-linear spectroscopy

1.1 Model HamiltonianTo model light-harvesting in photosynthetic systems, we use the Heitler-London modelHamiltonian [3, 93], written as a sum of terms including an electronic system (S), a vi-brational reservoir (R) and light (L):

H = HS +HS-R +HR +HS-L. (1.1)

We use a Frenkel exciton model for the system,

HS =∑n

Ena†nan +∑n 6=m

Jnma†nam (1.2)

where an is the annihilation operator for an electronic excitation on pigment n, En theexcitation energy on pigment n and Jnm = Jmn the dipole-dipole coupling between pigmentsn and m. We treat these excitations as hard-core bosons (that is, not allowing doubleexcitations of a single pigment), and restrict our consideration to only one possible excitationper pigment molecule. The excitation energies En are nominally identical for all pigments ofthe same type, but in practice are shifted significantly between different pigments becauseof differences in their local electrostatic environments.

For consideration of transport properties, it is sometimes convenient to restrict our con-sideration to exactly one excitation in the entire system. In such a case, we define the states|n〉 = a†n|g〉, where g denotes the electronic ground state of the entire system, and use themto write the system Hamiltonian restricted to the single excitation manifold,

H1-excitationS =

∑n

En|n〉〈n|+∑n 6=m

Jnm|n〉〈m|. (1.3)

CHAPTER 1. INTRODUCTION 2

We refer to these individual pigment states |n〉 as site states and the eigenstates of Eq. (1.3)as exciton states. The site basis refers to the basis of individual pigments in which theseequations are written; the exciton basis refers to the basis constructed from the exciton statesin which HS is diagonal.

The reservoir (bath) is modeled as a collection of harmonic oscillators initially at thermalequilibrium,

HR =∑ξ

~ωξb†ξbξ (1.4)

where bξ denotes a reservoir annihilation operator and ~ωξ the energy of an excitation inreservoir mode ξ. From a physical perspective, the reservoir represents a normal modeexpansion of all external degrees of freedom beyond the electronic state of each pigment,including internal intra-pigment vibrations, as well as motion of the surrounding protein andsolvent. The system-reservoir coupling is taken to be a sum of linear couplings between thetransition energy of pigments and the displacement of a bath mode,

HS-R =∑n,ξ

~ωξgnξa†nanqξ, (1.5)

where gnξ denotes the unit-less strength of the coupling between electronic excitation n

and bath mode ξ, and qξ = bξ + b†ξ is the unit-less displacement operator for mode ξ. Allinformation regarding the reservoir is contained in the spectral density function Jn(ω) =∑

ξ g2nξδ(ω − ωξ). Lastly, for far-field interactions betwen the system and applied fields, the

system-light interaction is given by

HS-L = µ ·E(t) (1.6)

where E(t) is the semi-classical electric field of the incident light and µ =∑

n dn(an + a†n)is the transition dipole operator, where dn is the transition dipole vector of pigment n.

The equations above describe the standard effective theory used for modeling energytransfer in light harvesting systems [93]. In the later chapters of this dissertation, we chooseparticular parameter values as determined by model fitting in prior experimental and the-oretical studies [2, 62, 94]. These values can, at least in principle, also be determined byab-initio quantum chemical calculations [97, 142]. However, my focus is on the features ofdynamics resulting from this model, not its experimental or microscopic derivation.

1.2 Energy transfer dynamicsBoth transport and optical properties can be described in terms of operators defined inthe electronic system. Hence, the main quantity of interest is usually the reduced densityoperator σ describing the electronic system. According to the usual terminology for densityoperators, we refer to the diagonal elements of density operators as populations and the


off-diagonal elements as coherences. The exact evolution of σ is given by a partial trace overthe reservoir states [12],

σ(t) = TrR[e−iHt/~ρ(0)eiHt/~], (1.7)

where ρ(0) denotes the initial state, usually taken to be the equilibrium density matrixe−βH/Tr[e−βH ], where β = 1/kBT is the inverse temperature. However, because the fullHamiltonian describes a many body quantum system, it is typically not feasible to useEq. (1.7) directly. When strictly the electronic Hamiltonian is considered, we see that theoff-diagonal elements 〈α|σ|β〉 in the energy eigenbasis oscillate like e−iωαβt, where ωαβ =(Eα−Eβ)/~. Such electronic quantum coherences, especially between one-excitation states,are a central focus of this work, because their presence indicates the possibility that quantumwave-like motion is relevant to transport in light-harvest harvesting systems.

The original approach to modeling excitonic energy transfer, known as Förster theory, isa perturbation theory expansion of all inter-pigment couplings Jnm to yield energy transferrates between pigment molecules of the form

kn→m =J2nm

2π~2

∫ ∞−∞

dωED(ω)IA(ω), (1.8)

where ED(ω) is the donor lineshape (for pigment n) and IA(ω) is the acceptor lineshape (forpigmentm) [78, 93]. The integral over the donor and acceptor lineshapes provides the densityof states for the transition between the deexcitement of pigment n and excitement of pigmentm according to Fermi’s golden rule. Although these lineshapes can be difficult to calculatefrom first principles, they are proportional to the donor emission and acceptor absorptionspectra [93], respectively, quantities that can be easily measured in the lab. The resultingdynamics, which in most cases constitutes a qualitatively accurate model of energy transferbetween pigments, are a classical random walk in continuous time with transition ratesgiven by Eq. (1.8). However, since Förster theory ignores coherences, it is not suitable forcalculations that require a full density matrix. Moreover, in cases where electronic quantumbeats or excitons delocalized between multiple pigments have been observed, Förster theoryis clearly outside of its realm of validity.

Under the approximation that the interactions between the system and the bath areMarkovian (memory less), we can prescribe the most general possible dynamics of a reduceddensity matrix such that it remains a valid. The result is that all such dynamics can bewritten in the form of the Lindblad master equation [12],

dσ

dt= − i

~[HS, σ] +

N2−1∑k=1

γk

(AkσA

†k −

1

2A†kAkσ −

1

2σA†kAk

), (1.9)

in terms of non-negative real valued constants γk and a set of operators Ak. Althoughthis Markovian approximation is not necessarily valid for energy transfer in light-harvesting


systems, models that take the form of Eq. (1.9) are highly useful because of their simple andcomputationally tractable form.

The simplest quantum model we use for energy transfer is the Haken-Strobl model [55],which takes the form of Eq. (1.9) with operators Ak = a†kak for each site k. In the literature ofopen quantum systems, this model is known as a pure dephasing model. The Haken-Stroblmodel can be derived under the assumption that each bath mode is actually a classicalGaussian random variable qξ(t) with a delta-function time-correlation 〈qξ(t)qξ(0)〉 ∝ δ(t),and that each mode is only coupled to one pigment. It can also be derived from a fullyquantum model of the reservoir in a high temperature limit [12, 118], in order to estimate thedephasing rates γk for each site in terms of the spectral density Jk(ω). The main advantage ofthe Haken-Strobl model is its ability to provide a simple transition from quantum to classicalmotion. The interpretation of the model is simple as well: the non-unitary contribution todynamics is simply an exponential decay of coherences between site states. In the long timelimit, one can show that the resulting density matrix for any finite system is diagonal andthat all site states n and m with Jnm 6= 0 have equal populations.

The Redfield model is a master equation model with a fully quantum microscopic deriva-tion. It requires the assumptions that the system-reservior coupling HS-R is weak and thatthe bath is Markovian. In such a case, the equilibrium density matrix can be factorized be-tween the system and bath, and a standard treatment using 2nd-order perturbation theory[93, 104] arrives at the Bloch-Redfield master equation,

dσabdt

=

(− i~

[HS, σ]

)ab

+∑cd

Rab,cdσcd. (1.10)

We sometimes call this equation the “full” Redfield theory, in contrast to the “secular” Red-field theory discussed below. The Redfield tensor Rab,cd, which is written in terms of thespectral density and terms in the system Hamiltonian, contains all dissipative dynamics.When written in the exciton basis, its terms are many be classified into three types:

1. a = b, c = d: Population transfer (a 6= c) and decay (a = c).

2. a = c, b = d, a 6= b: Coherence decay.

3. All others: Coherence to coherence, coherence to population and population to co-herence transfer terms.

The Redfield master equation cannot necessarily be written in the form of a Lindblad mas-ter equation, since the third type of these terms can lead to dynamics that do not preservethe positivity of the system density matrix. An additional approximation, the secular ap-proximation, neglects these terms, since unlike the population transfer/decay and coherencedecay terms, they oscillate with a non-zero frequency ωab − ωcd when the system densitymatrix is written in the interaction picture with respect to HS, and thus their contributionshould average out to zero. Because only the population transfer and coherence decay terms


remain, secular Redfield theory can be interpreted intuitively in the exciton basis, as a classi-cal random walk in continuous time with exponentially decaying coherences. A particularlyconvenient feature of secular Redfield theory is that, in contrast to the Haken-Strobl model,in the long time limit the state arrives at the correct equilibrium, σeq ∝ e−βHS .

Although Lindblad master equations are convenient, for most light-harvesting systems,models with a high temperature (classical) bath or weak system-bath coupling are not ac-curate. Indeed, natural light harvesting systems typically exhibit a convergence of energyscales, |En − Em| ∼ Jnm ∼ |HS-R| ∼ kBT ∼ ~/τR (τR denotes the time-scale of reservoirmotion). To meet the challenge of solving exact energy transfer dynamics, a variety of com-putationally expensive exact numerical methods have been developed [76, 122, 72, 146, 33].In some later chapters, we use a technique known as the 2nd-order cumulant time-nonlocal(2CTNL) quantum master equation or the hierarchical equations of motion (HEOM) ap-proach [76], which results in computationally tractable equations of motion for particularforms of the spectral density. This model is quantitatively accurate in the regimes of validityof both the Förster and Redfield approaches [76].

1.3 Non-linear optical spectroscopyWe can experimentally observe how energy moves through light harvesting systems withtime-resolved spectroscopy, that is, measurements of how they absorb and emit light. Be-cause energy transfer dynamics in natural light harvesting systems take place over femto- topicosecond timescales, ultrafast laser pulses are required for the necessary time resolution.

Experiments on natural light harvesting systems use weak laser fields, because strongfields, which can be defined as fields strong enough to excite more than a small fraction ofthe pigment molecules, quickly destroy most molecular light harvesting systems. Naturallythen, the theoretical formalism for these experiments is a perturbation theory in terms ofthe system-field interaction HS-L. The experimental signal field ES(t) is proportional to thenth order polarization, which is given by

P (n)(t) = Trµρ(n)(t), (1.11)

where ρ(n)(t) is the density matrix to nth order in HS-L. An additional important factor isthat each applied fields Ei(t) has an associated wave-vectors ki, so the final signal field musthave a wave-vector of the form kS =

∑i±ki, in terms of the positive or negative contributions

from each pulse. The particular choice of kS, as determined by the experimental geometry,thus allows for selecting particular experimental signals with this phase-matching condition.The ability to neglect components of ρ(n)(t) that oscillate in non-observed directions turnsout to considerably simplify the interpretation of experimental signals.

There are a wide range of possible non-linear spectroscopic measurements [98], deter-mined by the choice of signal orientation and polarization, in addition to the polarizations,orientations and time-profiles of all applied fields. In the last part of this dissertation, wefocus on third-order spectroscopy, the lowest order spectroscopy at which energy transfer


dynamics are directly observable. As has been shown elsewhere [1, 98], the phase-matchedcontribution to the third-order polarization can be written in the response function formalismas

P(3)ks

(t) =

∞y

0

dt3 dt2 dt1 R(3)ks

(t3, t2, t1)Eu33 (t− t3)Eu2

2 (t− t3 − t2)Eu11 (t− t3 − t2 − t1)

(1.12)

assuming that the electric fields Euii (t) interact with the system in the numbered order (as

guaranteed by a lack of temporal pulse overlap) and invoking the rotating wave approxi-mation, which is generally valid for resonant excitation (the case considered in this disser-tation). The rotating wave approximation is important for efficient calculations because itremoves extremely fast oscillations due to applied fields, which are ∼ 100 times faster thanenergy transfer dynamics. The quantity kS = u1k1 + u2k2 + u3k3 is the signal wave-vector,where (u1, u2, u3) ∈ {(−,+,+), (+,−,+), (+,+,−)} correspond to the three experimentalgeometries with non-zero signal (rephasing, non-rephasing and double-quantum-coherence,respectively) and E+

i (t) denotes the complex profile of the ith pulse (we use the conventionE− = (E+)∗). The system dynamics are contained in the phase-matched components of thethird order response function R(3)

ks(t3, t2, t1), which is given by

R(3)kS

(t3, t2, t1) =

(i

~

)3

Tr[µ(−)G(t3)V u3G(t2)V u2G(t1)V u1ρ0

], (1.13)

where we defined the annihilation portion of the dipole operator µ(−) =∑

n dnan and thecreation portion µ(+) = (µ(−))

†, the Liouville space operator G(t) is the retarded materialGreen function for evolution for time t and V (±) · ≡ [µ(±), ·].

Figure 1.1 illustrates the experimental geometries for a generic third-order spectroscopysetup (such the photon-echo) and for a pump-probe setup. Pump-probe is a special case ofthird-order spectroscopy that we focus on in later chapters, in which the first two interactionsare with the same pulse (k1 = k2), called the pump, and the third interaction is with theprobe field. The signal is observed in the direction kS = k3, so u1 = −u2. Accordingly,the signal is given by adding together the rephasing and non-rephasing interactions, andboth pulse orderings 1-2-3 and 2-1-3. Since light absorbed directly from the probe pulse (afirst-order signal) also appears in the signal direction, the third-order signal in the probedirection is calculated as the differential absorption in the probe direction with and withoutthe pump pulse.

This response function formalism neatly separates the system and field related quantities,and provides a benchmark method for calculating non-linear spectroscopy signals (directlyusing these equations). For the state tomography and control studies of pump-probe spec-troscopy discussed in Chapters 4 and 5, we use a more computationally efficient techniquethat we introduce in Chapter 4, and which we extend to simulating general third-order signalsin Section 5.7.2. These alternate simulation approaches use equations of motion explicitly


E1(t)

E2(t)

E3(t) ES(t)

k1

k2

k3

kS ES(t)

kpu

Epr(t)

Epu(t) kpr = kS

(a)! (b)!

Figure 1.1: (a) The setup for a generic third-order spectroscopy experiment. (b) The setupfor a pump-probe experiment.

including the system-light coupling HS-L in place of calculating the complete third-order re-sponse function. Since the third-order response function provides information sufficient tocalculate any concievable third-order experiment, these alternate approaches can sometimesresult in significant computational savings.

1.4 Flexible simulation of dynamics and spectroscopyThe diversity of methods for simulating both the dynamics and spectroscopy of light-harvestingsystems, as outlined in the preceeding sections, presents an interesting software engineeringchallenge, since it is impractical to individually code each possible combination of differentdynamics and spectroscopy methods. Accordingly, we have designed an application program-ming interface (API) that provides a sufficient interface to couple any method for efficientsimulating of energy transfer dynamics with any method for calculating a spectroscopic sig-nal. The API is described here and also implemented in a software package currently underdevelopment.

The core this API is the dynmical model interface, which specifies a sufficient set of oper-ations for manipulating the arbitrary state vectors x which describe the complete electronicand vibrational system under any model. For example, for a treatment with a Markovianbath, x could be a Liouville space representation of the system density operator. The in-terface consists of three parts. First, the initial condition x0 provides the overall state forthe system in the ground electronic state. Second, an equation of motion f(t,x) specifiesthe free evolution of the system in the absence of a probe field,

∂x

∂t= f(t,x). (1.14)

Finally, creation and annihilation dipole operators, are defined by objects obeying asystem-operator interface defined by the action (in term of the state vector) of left-multiplication


V ρ, right-multiplication ρV and expectation values Tr[V ρ] of arbitrary electronic operatorsV on states x equivalent to the density operator ρ.

To enable efficient calculations, the dynamical model interface also allows for followingonly particular Liouville space pathways [98]. These Liouville space pathways are importantbecause the only part of the Hamiltonian which changes the total number of electronicexcitations is the system-field coupling. In the absence of a system-field interaction, theright and left sides of the density matrix remain in fixed subspaces in terms of the number ofelectronic excitations. Because the dipole-operator is applied a fixed number of times in non-linear spectroscopy, we can explicitly map the possible pathways between states in differentsubspaces. Ignoring redundant or irrelevant subspaces allows for significant improvementsin computational efficiency, particularly because the number of distinct states with a fixednumber of excitations grows rapidly with the total number of sites. For example, if theground, 1-excitation and 2-excitation subspaces are denoted by the letters g, e and f , theonly states relevant to the calculation of a pump-probe signal are of the form |g〉〈g|, |g〉〈e|,|e〉〈g|, |e〉〈e| and |f〉〈e|. Accordingly, the interface allows for specifying the initial conditionand equation of motion restricted to any desired subspace, and we should be able to restrictthe action of the dipole operators to a fixed transition between subspaces.

9

Part I

Biological relevance of quantumcoherence

10

Chapter 2

Limits of quantum speedup inphotosynthetic light harvesting

SummaryIt has been suggested that excitation transport in photosynthetic light-harvesting complexesfeatures speedups analogous to those found in quantum algorithms. Here we compare thedynamics in these light-harvesting systems to the dynamics of quantum walks, in order toelucidate the limits of such quantum speedups. For the Fenna–Matthews–Olson complexof green sulfur bacteria, we show that while there is indeed speedup at short times, this isshort lived (70 fs) despite longer-lived (ps) quantum coherence. Remarkably, this timescaleis independent of the details of the decoherence model. More generally, we show that thedistinguishing features of light-harvesting complexes not only limit the extent of quantumspeedup but also reduce the rates of diffusive transport. These results suggest that quantumcoherent effects in biological systems are optimized for efficiency or robustness rather thanthe more elusive goal of quantum speedup.

2.1 IntroductionIn the initial stages of photosynthesis, energy collected from light is transferred across anetwork of chlorophyll molecules to reaction centers [9, 30]. Recent experimental evidenceshowing long lived quantum coherences in this energy transport in several photosyntheticlight-harvesting complexes suggests that coherence may play an important role in the func-tion of these systems [44, 86, 75, 21, 37, 108]. In particular, it has been hypothesized thatexcitation transport in such systems to feature speedups analogous to those found in quantumalgorithms [44, 96]. These comments have attracted much interest from quantum informa-tion theorists [96, 118, 117, 111, 26, 25, 34, 125, 152, 11], although clearly photosynthesis isnot implementing unitary quantum search [96]. The most direct analogy to such transportis found in quantum walks, which form the basis of a powerful class of quantum algorithms

CHAPTER 2. LIMITS OF QUANTUM SPEEDUP 11

including quantum search [5, 32, 141, 4, 6, 31]. Unlike idealized quantum walks, however, reallight harvesting complexes are characterized by disorder, energy funnels and decoherence.Whether any quantum speedup can be found in this situation has remained unclear.

Quantum walks are an important tool for quantum algorithms [5, 32, 141, 4, 6, 31]. On theline, quantum walks feature ballistic spreading, 〈x2〉 ∝ t2, compared to the diffusive spreadingof a classical random walk, 〈x2〉 ∝ t, where 〈x2〉 denotes the mean squared displacement.Moreover, because they use superpositions instead of classical mixtures of states, on anygraph with enough symmetry to be mapped to a line, quantum walks spread along that linein linear time – even when classical spreading is exponentially slow [32]. We shall refer tothis enhanced rate of spreading as a generic indicator of “quantum speedup.” Such quantumspeedup is important in quantum information processing, where it can lead to improvedscaling of quantum algorithms relative to their classical alternatives. Examples of suchalgorithmic speedup include spatially structured search [141], element distinctness [4] andevaluating and-or formulas [6]. Quantum walks even provide a universal implementationfor quantum computing [31].

Quantum walks also constitute one of the simplest models for quantum transport onarbitrary graphs. As such they provide a theoretical framework for several physical pro-cesses, including the transfer of electronic excitations in photosynthetic light harvestingcomplexes [44, 96]. The closed system dynamics in a light-harvesting complex are gener-ally well described by a tight-binding Hamiltonian that is restricted to the single excitationsubspace [30],

H =∑n

En|n〉〈n|+∑n6=m

Jnm|n〉〈m|. (2.1)

Here |n〉 represents the state where the nth chromophore (site) is in its electronic excitedstate and all other chromophores are in their ground state. En is the electronic transitionenergy of chromophore n and Jmn is the dipole-dipole coupling between chromophores nand m. This is a more general variant of the Hamiltonian for standard continuous-timequantum walks, where En = 0 and Jmn ∈ {0, 1}. The salient differences of the generalmodel, variable site energies and couplings, arise naturally from the structure and role oflight-harvesting complexes. Non-constant site energies can serve as energy funnels and canenable the complex to absorb at a broader range of frequencies, while variable couplingsbetween sites reflect their physical origin as dipole-dipole interactions. These differencesyield excitation dynamics that can deviate significantly from quantum walks.

Here we study the key question of whether excitation transport on light-harvesting com-plexes shows quantum speedup. Such quantum speedup would be necessary for any quan-tum algorithm that offers algorithmic speedup relative to a classical search of physical space.(Achieving true algorithmic speedup, i.e., improvement over the best classical algorithms,would also require suitable scaling of the space requirements [10].) We address this with astudy of the Fenna-Matthews-Olson (FMO) complex of green sulfur bacteria, a small andvery well-characterized photosynthetic complex [9, 30, 47, 2], the same complex for whichlong lived quantum coherences were recently observed and suggested to reflect execution of a


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Figure 2.1: (a) Crystal structure of FMO complex of C. tepidum (Protein Data Bank ac-cession 3ENI), with lines between the chromophores representing dipolar couplings. Thethickness of the lines indicates the coupling strengths. Only couplings above 15 cm−1 areshown; the largest coupling is 96 cm−1. The full Hamiltonian is given in 2.8.1. (b) Siteenergies, shown relative to 12 210 cm−1, in the reduced dimensionality model derived frommapping the strongest couplings onto a one-dimensional graph. The red sites (1, 6) aresource sites at which the excitation enters the complex and the black site (3) is the trap sitefrom which the excitation is transferred to the reaction center [2, 151].

natural quantum search [44]. We also consider the dynamics of transport along a theoreticalmodel of an extended chain of chromophores to elucidate the systematic influence of variablesite energies and dephasing on transport.

2.2 Fenna-Matthews-Olson complexThe FMO complex acts as a quantum wire, transporting excitations from a large, disorderedantennae complex to a reaction center. In addition to the possibility of quantum speedup [44,96], recent studies have speculated that coherence may assist unidirectional transport alongthis wire [75] or suggested that it may contribute to overall efficiency [117]. The crystalstructure of FMO shows three identical subunits that are believed to function independently,each with seven bacteriochlorophyll-a molecules embedded in a dynamic protein cage [47].A refined model Hamiltonian for the single excitation subspace is available from detailedquantum chemical calculations [2], and the orientation of the complex was recently verifiedexperimentally [151]. By neglecting the weakest couplings in this model Hamiltonian, we seethat transport in an individual monomer of FMO can be mapped to a one-dimensional pathbetween chromophores, as shown in Figure 2.1. This mapping of the excitation transportin FMO to a single dimension allows us to make contact with known results for quantumtransport in one dimension and to quantitatively assess the extent of quantum speedup.


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Figure 2.2: The spread of an initially localized wavepacket along a line for (a) the quantumwalk (no disorder), (b) under Stark localization (a linear gradient of site energies) showingthe characteristic Bloch oscillations, and (c) under Anderson localization (random variationof site energies).

2.3 Coherent dynamicsUnder Hamiltonian dynamics, quantum walks on highly symmetric graphs can spread bal-listically, but any significant lack of symmetry can lead to localization. Consider transportalong the infinite line. Here, a random variation of any magnitude in the site energies leadsto Anderson localization [110]. Similarly, random variations in the coupling strengths con-tribute to localization [48]. Systematic variations in site energies or coupling strengths canalso cause localization [138], with the well-known instance of Bloch oscillations and Stark lo-calization deriving from a linear bias in site energies [61]. Unitary transport without disorder,and under Anderson and Stark localization is illustrated in Fig. 2.2. Figure 2.3 highlightsthe particular case of Stark localization with ascending site energies Ek = kF by plotting


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Figure 2.3: (a) The energy eigenfunctions under Stark localization as given by Eq. (2.2) forJ/F = 5. (b) A position-energy plot of these energy eigenfunctions for a finite chain of 100sites. An excitation initially at the position shown by the dashed line only has finite overlapwith the states highlighted in green, and thus is locallized to this region.

the eigenvectors of the Hamiltonian, which are given by

|k〉 =∑n

Jn−k(2J/F )|n〉, (2.2)

where Jα(x) denotes the Bessel function of the first-kind of order α, and J is the constantcoupling between all adjacent sites. Both systematic and random variations in site energiesremove the symmetry necessary for Bloch’s theorem, so it is not surprising that their com-bination leads to localization as well [90, 81], as illustrated in Fig. 2.4. Adding disorder intothe graph structure also usually causes localization [100] (although such disorder can alsoreduce localization if it makes an already disordered graph more connected [52]). The variedcouplings, energy funnel and disordered energies evident in the FMO Hamiltonian depictedin Figure 2.1 suggest that localization due to several of these effects will be significant forlight-harvesting complexes. The standard measure of localization under coherent dynamics,the inverse participation ratio [110] ξ =

∑i |ψi|2/

∑i |ψi|4, for the amplitudes ψi in the site

basis of an eigenstate ψ of our model Hamiltonian confirms this intuition, since the typicaleigenstate for FMO occupies only ξ ∼ 2 sites (see also Ref. [14]).

We can estimate the timescale for localization tloc based upon the experimentally accessi-ble parameters of geometry, localization length ξ and an average coupling strength J . Con-sider the speed of the quantum walk as an upper bound on excitation transport speed priorto localization. For the continuous-time quantum walk on the infinite line, it is well knownthat 〈x2〉 ∼ 2J2t2/~2 [83]. Similarly, starting at one end of an infinite line, 〈x2〉 ∼ 3J2t2/~2.This gives an average speed gJ/~, where g =

√2 and g =

√3 respectively, yielding the

bound tloc & ~ξ/gJ . Our simulations of strict Hamiltonian dynamics (no decoherence) with


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Figure 2.4: Panels (a) and (b) show the spread of an initially localized state over time underthe two Hamiltonians with varying site-energies shown in (c) and (d). In panels (c) and (d),the energy-eigenfunctions are plotted in black, blue lines show the central energies En, andred lines shows the boundaries of the “forbidden zone” hypothesized by Ref. [138] as En±2J .

FMO as described below find the onset of localization at tloc ∼ 70 fs, which is close to thebound ~ξ/gJ ∼ 100 fs from the mean inverse participation ratio ξavg ≈ 2, the mean couplingstrength Javg ≈ 60 cm−1 and g =

√3 (this value for g is suggested by the dominant pathways

in Figure 2.1 for transport starting at sites 1 or 6, which are the chromophores closest to theantenna complex [2, 151]).

2.4 DecoherenceDecoherence, i.e. non-unitary quantum evolution, is also an essential feature of excitationdynamics in real systems. In light harvesting complexes decoherence arises from interactionswith the protein cage, the reaction center and the surrounding environment. Recently it has


been shown that some degree of fluctuation of electronic transition energies (i.e. dephasing inthe site basis) increases the efficiency of transport in FMO and other simple models otherwiselimited by Anderson localization. The intuition is that at low levels dephasing allows escapefrom localization by removing destructive interference, but at high levels it inhibits transportby inducing the quantum Zeno effect [96, 118, 111, 26].

We consider here two essential types of decoherence: dephasing in the site basis and loss ofexcitations. Under the Born–Markov approximation, time evolution with this decoherencemodel is determined by the system Hamiltonian H and a sum of Lindblad operators Laccording to [12]

∂ρ

∂t= − i

~[H, ρ] + Lloss(ρ) + Ldeph(ρ). (2.3)

Lloss describes loss of excitations with site dependent rates γn, including trapping to a reactioncenter, and is specified by the operator

〈n|Lloss(ρ)|m〉 = −γn + γm2

〈n|ρ|m〉, (2.4)

which results in exponential decay of the diagonal elements of the density matrix ρ (andcorresponding decay of off-diagonal elements, which ensures positivity). Ldeph describesdephasing, which is specified by the operator

〈n|Ldeph(ρ)|m〉 = −Γn + Γm2

(1− δnm)〈n|ρ|m〉 (2.5)

with site dependent dephasing rates Γn, and yields exponential decay of the off-diagonalelements of ρ.

A remarkable feature of our results for FMO is that, as we shall demonstrate, they areindependent of the finer details of the bath dynamics and system-bath coupling. Thus forsimplicity, we specialize here to the Haken-Strobl model [56] and restrict losses to trappingby the reaction center. The ease of calculations with the Haken-Strobl model has made itpopular for simulating the dynamics of light harvesting complexes [118, 111, 26, 88, 49].Following Rebentrost et al. [118], we use the spatially uniform, temperature dependent de-phasing rate Γ = 2πkBTER/~2ωc, for an ohmic spectral density with bath reorganizationenergy ER = 35 cm−1 and cut-off frequency ωc = 150 cm−1. This dephasing rate holdswhen chromophores are treated as qubits coupled to independent reservoirs in the spin-boson model, in the thermal (or Markovian) regime t � ~/kBT [12], which corresponds tot� 25 fs at 300K. Although the exact parameters of the protein environment surroundingFMO are unclear [75], these are reasonable estimates. This model gives dephasing rates(69 fs)−1 at 77K and (18 fs)−1 at 300K. Also, as in Ref. [118] we restrict trapping to site 3at a rate of γ3 = 1 ps−1. Finally, we neglect exciton recombination since it occurs on muchslower timescales (∼1 ns−1) and in any case it does not alter the mean-squared displace-ment. Therefore, we set γi = 0 for i 6= 3. While more realistic decoherence models wouldincorporate thermal relaxation, spatial and temporal correlations in the bath and strongsystem-bath coupling [76], this treatment is sufficient for analyzing the quantum speedup,


as justified in detail by comparison with the corresponding analysis using more realisticsimulations of FMO [76] in 2.8.2.

2.5 Limits of quantum speedupTo access the extent of quantum speedup for a system that can be mapped to a line, a naturalmeasure is the exponent b of the power law for the mean squared displacement 〈x2〉 ∝ tb.In particular, we use the best-fit exponent b from the slope of the log-log plot of the meansquared displacement 〈x2〉 = Tr[ρx2]/Tr ρ versus t. The value b = 1 corresponds to the limitof diffusive transport, whereas b = 2 corresponds to ideal quantum speedup as in a quantumwalk (ballistic transport).

The timescale for quantum speedup is generally bounded above by both the timescale fordephasing, tdeph = 1/Γ, and the timescale for static disorder to cause localization, tloc. To seethis, it is illustrative to consider transport along a linear chain with constant nearest neighborcouplings J . For dephasing but no static disorder or energy gradient (En = 0), the meansquared displacement shows a smooth transition from ballistic to diffusive transport [135,53] 1,

〈x2〉 =4J2

~2Γ

[t+

1

Γ

(1− e−Γt

)]. (2.6)

The corresponding power law transition is shown in Figure 2.5a. Technically, transportremains super-diffusive even after tdeph, since the power never drops below b = 1. In contrast,for static disorder but no dephasing, there is a sudden transition from ballistic transport toessentially no transport at all (localization). The power law starts at b = 2, then drops andbegins to oscillate wildly after tloc as the wave function continues to evolve in a confinedregion. Several examples are shown in Figure 2.5b for variable strengths of disorder. Figure2.5c shows the behavior with both strong static disorder and dephasing, as in light harvestingcomplexes. In this case, the transport can even exhibit a sub-diffusive power law. The reasonfor this sub-diffusive behavior may be easiest to understand by analogy to the Andersonmodel (random site energies) in an infinite chain. Consider transport in such a system underweak dephasing and with an ensemble average over different realizations of strong staticdisorder. In this case, one expects transport must transition from ballistic (b = 2) at shorttimes, to localized (b = 0) at intermediate times, to diffusive at long times (b = 1). (Thesituation at long times is analyzed more explicitly in Section 2.6 and 2.8.3.) For strongerdephasing as in Figure 2.5c (and as we shall see with FMO), the fully localized regime withno transport may never be realized, but transport will still be sub-diffusive for intermediatetimes.

Figure 2.6 shows the results of a simulation of FMO dynamics made with our simpledecoherence model. The displacement xi of a site i is given by its position in the one-dimensional mapping that is presented in Figure 2.1b. Note that our simulations use the

1This result, obtained in the context of excitation transport in molecular crystals, also applies to deco-herent continuous-time quantum walks since they use the same tight-binding Hamiltonian.


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Figure 2.5: The best fit power law exponent b for the mean-squared displacement withJ/~ = 1, for transport along a linear chain with either (a) increasing dephasing (Γ = 1, 3, 9)and no static disorder (En = 0), (b) no dephasing (Γ = 0) and increasing static disorder(independent site energies En from a single Gaussian distribution normalized to standarddeviation σ = 1, 3, 9) or (c) same disorder as (b), but with finite dephasing rate Γ = 1.Panels (b) and (c) show results for single instances of disorder. When an ensemble averageover different realizations of static disorder is taken, the power law varies smoothly insteadof oscillating as in (b) and (c).


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Figure 2.6: Results of simulations on FMO with the decoherence model described in thetext. (a) Log-log plot of mean-squared displacement 〈x2〉 as a function of time, for an initialexcitation at site 6. (b) Power law exponent for the mean-squared displacement, given bythe slope of the plot in panel a. The dashed line b = 1 separates super- and sub-diffusivetransport (see text). (c) Total site coherence, given by the sum of the absolute value of alloff-diagonal elements of the density matrix in the site basis, indicates persistent coherencefor ∼ 500 fs. (d) Oscillating site populations (shown for 77K only) also indicate persistentcoherence for hundreds of femtoseconds.

full Hamiltonian; the one-dimensional map is only used in the analysis of the results, todetermine the displacement xi of each site. The upper-left panel, (a), shows the mean squareddisplacement 〈x2〉 of an excitation initially localized on site 6, one of the two excitationsource sites, and the lower-left panel, (b), the best fit exponent b for the power law 〈x2〉 ∝ tb.Our results show that even though coherence in this simple model lasts for ∼ 500 fs (seeFigure 2.6c), a transition from initially ballistic to sub-diffusive transport occurs after only∼ 70 fs, independent of the dephasing rate. While increased dephasing causes a faster initialdecrease of the power law, dephasing alone cannot lead to sub-diffusive transport, as is clear


from Equation (2.6). We note that the transition to sub-diffusive transport at 70 fs occursat the same time-scale as the onset of localization (Section 2.3), implying that even thoughthere is persistent coherence beyond this time, it no longer yields quantum speedup becauseof static disorder. The sub-diffusive power law at intermediate times (∼ 100 fs – 2 ps) arisesfrom the interplay of this disorder-induced localization and dephasing that is discussed abovefor the linear chain model. At longer times the power law exponent b goes to zero because ofthe finite size of the system. Since complete energy transfer through FMO takes picoseconds,this analysis shows that most of the excitation transport is formally sub-diffusive.

While FMO is a relatively small system, so that terms such as “ballistic” and “diffusive”cannot literally describe transport across its seven chromophores, this time dependence ofthe power law of spreading would also characterize larger artificial or natural systems. Ourresults here are robust to variations in the strength of the trap, and to whether the initialexcitation is at site 1 or 6, the sites believed to be the primary source for excitations inFMO [2]. They are also independent to variations of the map to a one-dimensional system inFigure 2.1, such as counting the number of chromophores away from the trap site instead ofalong the entire line, or using the real space distance between adjacent chromophores insteadof assuming that each pair is merely separated by a unit lattice distance.

Our timescale for short-lived quantum speedup in FMO should be largely independentof the details of the decoherence model, since the non-uniform energy landscape will alwayslimit ballistic transport to times prior to tloc ∼ 70 fs. This is confirmed by applying ouranalysis to the results of recent calculations for FMO made with a considerably more sophis-ticated decoherence model that incorporates thermal relaxation, temporal correlations in thebath and strong system-bath coupling [76]. As demonstrated in 2.8.2, these more realisticcalculations also predict a loss of quantum speedup after 70 fs.

2.6 Diffusive transportTo gain general insight into the interplay between disorder and dephasing for light harvestingcomplexes, we now consider the dynamics of an infinite linear chain at long times withvariable site energies, couplings and dephasing rates. By extending an analysis for dampedBloch oscillations [82], we find that any set of non-zero dephasing rates Γn asymptoticallyleads to diffusive transport 〈x2〉 ∼ 2Dt. Haken and Reineker also performed a reductionof the Haken-Strobl model without disorder to a diffusion equation along similar lines [54].Note that since the Haken-Strobl model does not include energy relaxation, no classical driftvelocity will be obtained from these dynamics. The time evolution of an arbitrary densitymatrix element for an infinite linear chain under site dependent dephasing rates is given by

ρnm = − i~

(Jn−1ρn−1,m + Jnρn+1,m − Jmρn,m+1 − Jm−1ρn,m−1)

−[i

~(En − Em) +

Γn + Γm2

(1− δnm)

]ρn,m, (2.7)


where Jn ≡ Jn,n+1. If we neglect terms in the second off-diagonal relative to the diagonalterms (see justification in 2.8.3) we obtain

ρn+1,n = − i~Jn(ρn,n − ρn+1,n+1)− (

i

~∆n + Γ′n)ρn+1,n, (2.8)

where ∆n ≡ En+1 − En and Γ′n ≡ (Γn + Γn+1)/2. This equation has the solution

ρn+1,n =i

~e−(i∆n/~+Γn)t

∫ t

0

Jn(ρn+1,n+1 − ρn,n)e(i∆n/~+Γn)t′dt′

≈ Jn(ρn+1,n+1 − ρn,n)

∆n − i~Γ′n, (2.9)

in the nearly stationary regime where site populations vary slowly compared to 1/Γ′n and~/∆n. Inserting (2.9) and the corresponding result for ρn−1,n into (2.7) we obtain

ρnn =2J2

nΓ′n∆2n + ~2Γ′2n

(ρn+1,n+1 − ρnn) +2J2

n−1Γ′n−1

∆2n−1 + ~2Γ′2n−1

(ρn−1,n−1 − ρnn). (2.10)

This is now a classical random walk with variable bond strengths between sites. For asymp-totically large times it has been proven to approximate the diffusion equation with coefficientgiven by

D =

⟨∆2n + ~2Γ′n

2

2J2nΓ′n

⟩−1

, (2.11)

as long as the average over sites in the diffusion coefficient is well-defined [71]. With ∆n, Jnand Γn constant, this matches the known diffusion coefficient [82, 41]. For constant couplingJn = J and constant dephasing Γ′n = Γ, we note that (2.11) reduces to

D =2J2Γ

〈∆2n〉+ ~2Γ2

, (2.12)

which is plotted as a function of 〈∆2n〉1/2 and Γ in Figure 2.7 2. Note that with 〈∆2

n〉 = 0,this is the long time limit of (2.6). In the case of Equation (2.12), for a given degree of staticdisorder, we find an optimal dephasing rate ~Γ = 〈∆2

n〉1/2. However, Figure 2.7 shows thatfor a given dephasing rate, excitation transport would be improved by reducing the staticdisorder to further delocalize the system.

2.7 ConclusionsThese principles show that quantum speedup across a photosynthetic or engineered systemrequires not only long-lived quantum coherences but also excitons delocalized over the entire

2We note that the parameter 〈∆2n〉1/2 does not completely specify the degree of localization, although it

does show that Stark and Anderson localization equivalently influence diffusive transport. For example, theinverse participation ratio in these two cases differs, even with the same value of 〈∆2

n〉1/2.


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Figure 2.7: Diffusion coefficients at long times as a function of a constant dephasing rate Γand the static disorder between adjacent sites, as calculated by (2.12) with J = ~ = 1. Solidlines are contours; the dashed line is the optimal dephasing rate for given static disorder.The ideal quantum walk, which is non-diffusive, is at the origin.

complex. Such completely delocalized excitons do not exist in FMO and are also unlikely inother light harvesting complexes because of the presence of energy gradients and disorder.Moreover, Equation (2.12) makes it clear that the conditions needed for longer lived quan-tum speedup (reduced dephasing and static disorder) are those necessary for faster diffusivetransport as well.

The short-lived nature of quantum speedup in light harvesting complexes that we have es-tablished here implies that the natural process of energy transfer across these complexes doesnot correspond to a quantum search. Indeed, neither a formal quantum nor classical searchmay be necessary, since pre-determined (evolved) energy gradients can guide relaxation toreaction centers, even though such gradients suppress coherent and dephasing-assisted trans-port. This is particularly relevant for systems such as FMO which either receive excitationsone at a time or have isolated reaction centers. Instead of yielding dynamical speedup likethat in quantum walk algorithms, quantum coherence in photosynthetic light harvesting ap-pears more likely to contribute to other aspects of transport, such as overall efficiency orrobustness. We emphasize that a restricted extent of quantum speedup does not imply thatthere is no significant quantum advantage due to long-lived coherence in electronic excita-tion energy transfer. Identifying the specific nature of any “quantum advantage” for FMOwill clearly require more detailed analysis of the dynamics, particularly in the sub-diffusiveregime. Related examples of such “quantum advantage" are found in the LH2 complex ofpurple bacteria, where coherence has been specifically shown to improve both the speed [78]and robustness [28] of transport from the B800 to B850 ring.


Note added

While revising this manuscript, a reduction of the Haken-Strobl model to a classical randomwalk similar to the one we perform in Section 2.6 was published by Cao and Silbey [23].

2.8 Appendices

2.8.1 FMO Hamiltonian

In all of our calculations, we use the Hamiltonian calculated for C. tepidum by Adolphs andRenger [2]. In the site basis, this is given by

HFMO =

200 −96 5 −4.4 4.7 −12.6 −6.2−96 320 33.1 6.8 4.5 7.4 −0.3

5 33.1 0 −51.1 0.8 −8.4 7.6−4.4 6.8 −51.1 110 −76.6 −14.2 −674.7 4.5 0.8 −76.6 270 78.3 −0.1−12.6 7.4 −8.4 −14.2 78.3 420 38.3−6.2 −0.3 7.6 −67 −0.1 38.3 230

, (2.13)

with units of cm−1 and a total offset of 12 210 cm−1. Bold entries indicate those shown inFigure 2.1. In units with ~ = 1, we note that the rate 1 ps−1 ≡ 5.3 cm−1.

2.8.2 Reduced hierarchy equations model for FMO

Energy transfer dynamics in photosynthetic complexes can be difficult to model becauseperturbations from the surrounding protein environment can be large, and the timescale ofthe protein dynamics is similar to the timescales of excitation transport. This makes commonapproximations involving perturbative treatments of system-bath coupling and Markovianassumptions on the bath invalid. Recently a non-perturbative, non-Markovian treatment ofenergy transfer has been formulated by Ishizaki and Fleming [76]. This model assumes: (i)a bilinear exciton-phonon coupling, (ii) protein fluctuations that are described by Gaussianprocesses, (iii) a factorizable initial state of chromophores and protein environment, (iv)protein fluctuations that are exponentially correlated in time, and (v) no spatial correlationsof fluctuations.

To verify our results in Figure 2.6 of the main paper with this more realistic model forexcitation dynamics, we apply the power law analysis of mean squared displacement to theresults of the simulations that were used to calculate entanglement dynamics in Ref. [125]using the non-perturbative non-Markovian method of Ref. [75]. These simulations used areorganization energy of the protein environment of 35 cm−1, phonon relaxation time of 100 fs,and a reaction center trapping rate of (4 ps)−1, all of which are consistent with literatureon FMO [2, 35]. Figure 2.8 shows the power law analysis for 2 different temperatures, 77K


10 100 1000 50000.0

0.5

1.0

1.5

2.0

Time HfsL

Pow

erla

w

77 K

300 K

Figure 2.8: For the reduced hierarchy equation model as applied to FMO, we plot the powerlaw for the mean squared displacement as in Figure 2.6b. The plot is interpolated from afourth order polynomial fit of density matrix elements for results from a numerical simulationon FMO with sampling approximately every 8 fs, with the initial excitation at site 6 [75].

and 300K. Our analysis of these realistic simulations show that longer lasting coherences(as discussed in Refs. [75, 76]) are evident in the power law oscillations, but that transportis still nevertheless sub-diffusive after ∼ 70 fs, for both temperatures.

It is particularly noteworthy that our results for FMO hold even under a model incorpo-rating finite temperature relaxation. In principle, thermal relaxation could bias the dynamicson a linear chain with a classical drift velocity [113] to give the power law 〈x2〉 ∝ t2 evenwithout quantum speedup. However, the above analysis of FMO simulations made with themost realistic treatment of relaxation available today (Figure 2.8) shows that this is not thecase in this system.

2.8.3 Decay of coherences in linear transport

For a constant dephasing rate Γ � J/~, there is a simple analytical argument that the off-diagonal elements of the density matrix in the site basis can be neglected relative to the maindiagonal [45]. In the case of transport along a line, application of the analysis of Ref. [45]to Equation (2.7) shows that the first off-diagonal elements decay at rate Γ, unless morecoherence is generated from the main diagonal. If the main diagonal elements have order1, then the first off-diagonal elements cannot have magnitude greater than order J/~Γ. Asimilar argument bounds the second diagonal elements as less than (J/~Γ)2 and so forth.For large Γ, this justifies neglecting this second off-diagonal relative to the main diagonal.The higher order terms in this expansion can also be calculated explicitly [23].

From extensive numerical experiments with infinite chains, we have found uniformly that


a

0.01 0.1 1 100.01

0.1

1

Time

Sum

ofk

thof

f-di

agon

al

Increasing k

b

0.01 0.1 1 10

Time

Increasing k

c

0.01 0.1 1 10

Time

Increasing k

Figure 2.9: Decay of coherences. Log-log plot of the sum of the absolute values kth off-diagonal elements of the density matrix over time for constant dephasing and couplingstrengths Γ = J/~ = 1. Panels show the quantum walk with dephasing (a), and Stark(b) and Anderson (c) localization with dephasing and 〈∆2

n〉1/2 = π/2. Note that the maindiagonal always sums to 1.

the off-diagonal elements of the density matrix decay nearly monotonically after time 1/Γeven when Γ � J/~ does not hold, although an analytic proof of this result has eluded us.Several examples are presented in Figure 2.9. This matches the analytical result for finitesystems in the Haken-Strobl model that coherence must vanish eventually for any non-zeroΓ [109].

26

Chapter 3

Spatial propagation of excitoniccoherence enables ratcheted energytransfer

SummaryExperimental evidence shows that a variety of photosynthetic systems can preserve quantumbeats in the process of electronic energy transfer, even at room temperature. However,whether this quantum coherence arises in vivo and whether it has any biological functionhave remained unclear. Here we present a theoretical model that suggests that the creationand recreation of coherence under natural conditions is ubiquitous. Our model allows us totheoretically demonstrate a mechanism for a ratchet effect enabled by quantum coherence,in a design inspired by an energy transfer pathway in the Fenna-Matthews-Olson complexof the green sulfur bacteria. This suggests a possible biological role for coherent oscillationsin spatially directing energy transfer. Our results emphasize the importance of analyzinglong-range energy transfer in terms of transfer between intercomplex coupling states ratherthan between site or exciton states.

3.1 IntroductionMounting experimental evidence for electronic quantum coherence in photosynthetic energytransfer [126, 44, 37, 108, 21] has spawned much debate about both the detailed natureand the biological role of such quantum dynamical features. Quantum coherence is usuallyencountered in the first, light harvesting stage of photosynthesis. It includes two distinctbut not mutually excludeusive phenomena that can be differentiated by the choice of basisused to describe the electronic excitations. In the site basis, corresponding to the excitationof individual pigment molecules, coherence emerges in molecular aggregates even in thermalequilibrium, since eigenstates are delocalized over multiple chromophores. Such coherence

CHAPTER 3. SPATIAL PROPAGATION OF EXCITONIC COHERENCE 27

between sites can enhance the rate of biological energy transfer by up to an order of magni-tude [133, 144, 78, 66]. In contrast, it is coherence in the exciton basis, that is, superpositionsof the energy eigenstates, which drives quantum beating via the Schrödinger equation. It isthis type of coherence on which we will focus here, and thus in the remainder of this paper,the term ‘coherence’ refers to coherence in the exciton basis. Photosynthetic systems atambient temperatures have been shown to exhibit this kind of quantum beating when arti-ficially excited [108, 37], but the significance of these discoveries remains unclear. A broaddeficiency is the lack of plausible physical mechanisms for how this coherence could arise inand influence biological energy transfer. For example, the suggestion that transport in thesesystems features speedups reminiscent of quantum search algorithms [44] has been shownto be invalid [68, 96]. Experimental observations of long-lasting and delocalized quantumbeats alone are not sufficient to determine that they are biologically relevant, since thesefeatures arise due to strong inter-chromophoric couplings [129], which independently yieldfast transport rates under almost any theoretical model, even those which entirely neglectquantum coherence.

In this work we address the question of what physical mechanisms lie behind the originand biological role of electronic coherence in the exciton basis. While our theoretical anal-ysis is general, to make illustrative demonstrations of specific mechanistic features we takephysical parameters from a prototypical system, the Fenna-Matthews-Olson (FMO) complexof green sulfur bacteria. FMO is an extensively studied protein-pigment complex [2, 9, 47]that exhibits excitonic quantum beats [108] and entanglement [125] at room temperature.Biologically, FMO acts as an energy transmitting wire, delivering an electronic excitationcreated by photon absorption in the chlorosome antenna to a reaction center where it in-duces charge separation. It exists in the form of a trimer with a protein backbone and 3× 7bacteriochlorophyll-a molecules, each with different transition energies set by the local elec-trostatic environment. These pigments and energy transfer pathways are illustrated in Figure3.1. Several quantitative estimates of the importance of excitonic coherence under particularmodels suggest that it makes ∼10% contribution to transfer energy transfer efficiency in thissystem [117, 154]. The section of the energy transfer path from site 1 to 2 is particularlyunusual, since it is energetically uphill while these sites also have the strongest electroniccoupling of any pair of sites in the complex. It has been speculated that these factors mayindicate a role for quantum coherence in contributing to unidirectional energy flow throughthis system by avoiding trapping in local minima of the energy landscape [75]. All other stepsof the FMO electronic energy transfer pathways in the direction towards the reaction centerare energetically downhill, consistent with a pathway that would be optimal for classicalenergy transport. A key question that is of special importance for the FMO complex, isthus how the system efficiently directs energy transport away from the chlorosome antennatowards the reaction center. Clearly the overall energy gradient in the system plays a role,but does electronic coherence also facilitate unidirectional energy transfer? More generally,can excitonic coherence assist excitation transfer over the uphill steps found in rough energylandscapes?

A striking feature of all experiments showing electronic quantum beatings in photosyn-


81 2

3

6

54

7

(a)

12200

12300

12400

12500

12600

Site

ener

gyHcm

-1

L

HbLBChl 8

BChl 1

BChl 2

BChl 3

BChl 4

Figure 3.1: Energy transfer pathways in a monomer of the FMO complex of Chlorobaculumtepidum (C. tepidum). (a) Side view of a monomer of the FMO complex [47], showing theprimary energy transfer pathways [14] toward the reaction center via site 3 and the inter-complex coupling (ICC) basis states that couple site 8 to the remainder of the complex. Siteoccupation probabilities for the ICC basis states are proportional to the area of the coloredcircles. (b) Site energies along the upper energy transfer pathway depicted in panel (a), withthe energy of site 8 approximated by the antenna baseplate energy [75]. Lines between sitesindicate weak (dashed, 10 cm−1 < J < 40 cm−1) and strong (solid, J > 40 cm−1) electroniccouplings. Room temperature is approximately 200 cm−1.

thesis to date is that they have been performed on small sub-units of light harvesting antennasystems, such as the 7 pigments of the FMO complex [126, 44, 108] or the 14 pigments inLHC II [21, 129]. Yet natural light harvesting antennas are typically composed of hundredsor thousands of pigment molecules organized into many pigment-protein complexes throughwhich energy passes on route to the reaction center [9]. In addition, natural excitation is bysunlight, not ultrafast laser pulses. There is dispute about whether or not coherences canarise after excitation by natural light [30, 92, 73, 19], but many pigment-protein sub-units areactually more likely to receive excitations indirectly, as a result of weak coupling to anothercomplex in the larger network. Clearly, understanding the role of excitonic coherence in asingle protein-pigment complex requires placing the energy transfer within and through thatsystem in the context of appropriate initial conditions, as determined by its role in the larger“supercomplex.” Accordingly, a second key question for evaluating the relevance of quantumbeats to energy transfer on biological scales is whether or not excitonic coherence could beeither arise or be maintained in the process of transfer between different sub-units.

In the remainder of this paper we shall address these two open questions with a generaltheoretical framework employing a novel basis for analyzing the excitonic dynamics of weaklycoupled pigment-protein complexes. First, we develop an analysis of such weakly coupledcomplexes that suggests a mechanism for how coherence should arise and recur in the processof energy transfer. We find that coherence can continue to be regenerated during long-rangeenergy transport between weakly linked sub-units of a larger excitonic system. This is


the ‘propagation’ of quantum coherence, whereby a process of continual renewal followingincoherent quantum jumps may allow non-zero coherence to last indefinitely, despite rapiddecay after each jump. This has significant implications for long-range energy transfer in lightharvesting supercomplexes composed of multiple units that individually support coherence,such as photosystems I and II [9]. Second, we address the question as to whether suchspatially propagated intra-complex quantum coherence enables unidirectional flow of energy,with a specific example that is inspired by the uphill energetic step in FMO. We construct anexplicitly solvable ratchet model to show that in this situation, the non-equilibrium natureof even limited quantum beating may allow for qualitatively new types of dynamics. Inparticular, we show that under biologically plausible conditions, these dynamical featurescould allow for the operation of a coherently enabled ratchet effect to enhance directed energyflow through light harvesting systems. These analyses provide new understanding of physicalmechanisms reliant on quantum coherence that could be relevant to the function of naturalphotosynthetic systems.

3.2 Spatial propagation of coherenceIn photosynthetic energy transfer, excitons typically need to travel through a series of protein-pigment complexes before reaching reaction centers [9]. To accurately understand the role ofcoherent dynamics between individual sub-units in such a supercomplex, it is first necessaryto understand which particular sub-unit states donate or accept excitations for inter-complextransfer. As we shall show in this work, the nature of these states informs us whether or notcoherence arises under natural conditions in the process of energy transfer. Moreover, in thecontext of a light-harvesting complex which is a subcomponent of a larger light-harvestingapparatus, the precise nature of the acceptor states on the complex and the donor statesfrom the complex is paramount to assessing the possible relevance of coherent dynamics inthe complex. To draw an analogy to the circuit model of quantum computation [103], thesestates serve as effective choices of initial states and measurement basis states, respectively,for dynamics on an individual complex. Both of these states need to differ from energyeigenstates in order for strictly unitary dynamics to influence measurement outcomes. Themeasurement outcomes correspond to observable energy transfer, for which differences inrates or success probabilities could in turn influence biological function.

Since inter-complex couplings are relatively weak, in our analysis we treat them per-turbatively, as in multichromophoric generalizations of Förster theory used to calculateoverall transition rates between donor and acceptor complexes consisting of multiple chro-mophores [78, 144, 133]. Our starting point is the equation of motion for the reduceddensity matrix, which is derived with the following adaptation of the multichromophoricenergy transfer rate model [78]. The zeroth order Hamiltonian is H0 = HD + HA whereHD = He

D +∑

ij BDij |Di〉〈Dj|+HgD, with H

eD the electronic Hamiltonian of the donor com-

plex, and corresponding definitions for the acceptor complex A. States |Dj〉 and |Ak〉 forj = 1, . . . , n and k = 1, . . . ,m form an arbitrary orthonormal basis for donor and acceptor


single-excitation electronic states and BDij are bath operators that couple the electronic chro-mophore states to environmental states of the pigment-chromophore system. The groundstate donor (acceptor) bath Hamiltonian Hg

D (HgA) can be taken without loss of generality to

be a set of independent harmonic oscillators. We assume that no bath modes are coupled toboth the donor and acceptor so that [HD, HA] = 0 [78]. The donor and acceptor complexesare coupled by a dipolar interaction Hc = J + J † with J =

∑jk Jjk|Dj〉〈Ak|. Calculating

the evolution of the reduced system density matrix σ = TrB ρ to second order in Hc ([78]and Appendix 3.7.1) yields

σkk′

dt=∑jj′k′′

Jj′k′′

4π~2

∫ ∞−∞

dω[Jjk E

j′jD (t, ω) Ik

′′k′

A (ω) + Jjk′ Ejj′

D (t, ω) Ikk′′

A (ω)]

(3.1)

dσjj′

dt= −

∑kk′j′′

Jj′′k′

4π~2

∫ ∞−∞

dω[Jjk E

j′j′′

D (t, ω) Ikk′

A (ω) + Jj′k Ej′′j′

D (t, ω) Ik′k

A (ω)]

(3.2)

for the reduced acceptor and donor density matrix elements, respectively, where ED(t, ω)and IA(ω) denote matrices of donor and acceptor lineshape functions (see Eqs. (3.13–3.14)).We emphasize that these results hold for arbitrary system-bath coupling strength, providedthat the donor-acceptor coupling is weak: at this point in our analysis we have not yet madeany assumption of weak system-bath coupling.

Instead of focusing on the multichromophoric energy transfer rate between complexesthat results from summing Eq. (3.1) or (3.2) over all diagonal terms [78], we focus hereon important features relevant to quantum coherence apparent from Eqs. (3.1–3.2) directly.These equations indicate that acceptor populations |Ak〉〈Ak| will grow and donor populations|Dj〉〈Dj| will decay only if there is at least one non-zero coupling term Jjk = 〈Dj|J |Ak〉 tothose states. Accordingly, we argue that the transfer of electronic states is most sensiblydescribed by the “inter-complex coupling” basis in which J is diagonal, rather than thesite or exciton (energy) basis, as is assumed in both the original and generalized [144, 133]Förster theories. This inter-complex coupling (ICC) basis is given by the singular valuedecomposition J = UDJU

†A, where J is a rectangular diagonal matrix and UD and UA

are unitary transformations of donor and acceptor electronic states. We can thus writethe inter-complex coupling as Hc =

∑l Jl(|Dl〉〈Al| + |Al〉〈Dl|) in terms of the ICC states

|Dl〉 = UD|Dl〉 and |Al〉 = UA|Al〉 for l ∈ {1, . . . ,min(n,m)}. In the ICC basis, the fullelectronic Hamiltonian in block-matrix form is

He =

[U †DH

eDUD U †DJUA

U †AJ†UD U †AH

eAUA

]. (3.3)

Since in general the transformation that diagonalizes J will not coincide with the (exci-ton) eigenbases of He

D and HeA, population growth of an acceptor ICC state |Al〉〈Al| thus

corresponds to growth of excitonic coherences.Although in principle Eqs. (3.1–3.2) specify all dynamics relevant to inter-complex trans-

fer, the time-dependent donor lineshape ED(t, ω) obscures the specific dependence on donor


density matrix elements. Accordingly, we also derive a time-convolutionless quantum masterequation (Appendix 3.7.2) under the additional assumption of weak coupling to the bathrelative to the donor electronic Hamiltonian He

D [12]. Under this approximation, we seethat growth of an acceptor population |Al〉〈Al| is proportional to populations only of thecoupled donor |Dl〉〈Dl|. Likewise, decay of a donor population |Dl〉〈Dl| is proportional topopulations only of that donor itself (see Eqs. (3.22–3.23) in Appendix 3.7.2). Accordingly,inter-complex transfer rates may show oscillations reflecting donor quantum beats, since theICC states on the donor which transmit excitations do not necessarily correspond to energyeigenstates. While this part of our argument is only rigorous in the case of weak system-bathcoupling, which is not necessarily the case for FMO and other light harvesting systems [75],our simulations find excellent agreement even for moderate strength environmental coupling,as we show below.

To test this analysis of inter-complex energy transfer, we first consider its predictionsfor the special case in which there is only one non-zero inter-complex coupling in the ICCbasis, Hc = J∗|D∗〉〈A∗|+h.c. If the acceptor is always initialized in the state ρ∗A = |A∗〉〈A∗|,then when back-transfer to the donor is neglected as is valid in the perturbative limit, theacceptor density matrix should be well described by

ρA(t) =

∫ t

0

dt′dpA(t′)

dt′G(t− t′)ρ∗A, (3.4)

where G(t) is the Greens function denoting evolution of the acceptor-bath system for timet, with the bath initialized at equilibrium. If the predicted donor state is correct, thenneglecting temporary bath reorganization effects, the rate of inter-complex transfer shouldthen be proportional to the population of the predicted donor state, for a predicted inter-complex transfer rate

dpA(t)

dt∝ pD∗(t), (3.5)

where pD∗(t) denotes the probability of the donor being in the state |D∗〉 and pA the totalprobability of the excitation being on the acceptor.

For a model system, these predictions show remarkable agreement with results derivedfrom an independent simulation based on a 2nd-order cumulant time-nonlocal (2CTNL)quantum master equation [76]. We consider transfer between two dimer complexes (labelledsites 1, 2 and sites 3, 4), with intra-dimer Hamiltonian parameters matching those of the1-2 dimer of FMO (Appendix 3.7.5, Eq. (3.43)) and inter-complex coupling J = J∗|2〉〈3|.We perform calculations in the limit J∗ → 0 (see Appendix 3.7.4), to ensure accuracy of theperturbative description and eliminate back-transfer effects. The 2CTNL calculations arecarried out at 300K with a bath modeled by a Debye spectral density with reorganizationenergy 35 cm−1 and correlation time 50 fs, with the initial condition on site 1 [75]. Figure 3.2compares simulated 2CTNL results with the predicted time-dependent inter-complex transferrate, Eq. (3.5), and acceptor density matrix, Eq. (3.4) (normalized to unity for greater


0.0

0.2

0.4

0.6

0.8

1.0

Tra

nsfe

rra

teHar

b.un

itsL

HaL

0 100 200 300 400 500Time HfsL

0.0

0.2

0.4

0.6

0.8

1.0

Acc

epto

rco

here

nce

2ÈΡΑ

ΒÈ

HbL

0.5

0.6

0.7

0.8

0.9

1.0

Ρ33

HcL

0.0

0.1

0.2

0.3

Re

Ρ34

HdL

0 100 200 300 400 500Time HfsL

-0.4

-0.3

-0.2

-0.1

0.0

ImΡ

34

HeL

Figure 3.2: Testing the theory of propagation of coherence via the inter-complex coupling(ICC) basis. Simulations were made for the coupled dimer model described in the text, withICC donor and acceptor states |D∗〉 = |2〉, |A∗〉 = |3〉, respectively, and initial condition|ψ0〉 = |1〉. (a) Simulated (solid black, 2CTNL) and predicted (dashed blue, Eq. (3.5))inter-complex transfer rate as a function of time. (b) Coherence between the two acceptorexcitonic eigenstates α and β as a function of time. (c) Population of site 3 in the acceptor,i.e., ρ∗A = |3〉〈3|, as a function of time. (d) Real and (e) imaginary parts of the 3-4 sitecoherence for the simulated (solid black) and predicted (dashed blue, Eq. (3.4)) acceptordensity matrix ρA, as a function of time. The acceptor density matrix ρA was normalized tounit probability at all times in panels (b-e).


clarity), calculated from the 2CTNL results. The results show that estimates based on thedominant elements of the ICC basis provide an accurate representation of both the energytransfer rate (panel a) and acceptor density matrix (panels b-e). We see that transfer ofexcitation in the ICC basis from |D∗〉 to |A∗〉 produces a superposition of acceptor eigenstates(of HA) that gives rise to excitonic coherence (panel b) and hence to oscillatory behavior ofboth the site populations (panel c) and coherences (panels d-e). Two features are of particularsignificance, since they show that these ICC-dominated dynamics satisfy the conditions thatare necessary for intra-dimer coherence to be relevant to larger scale energy transfer, namelythat the dynamics guarantee the preparation and measurement of states which are notenergy eigenstates. The first feature is that the inter-dimer transfer rate clearly trackscoherent oscillations of the donor population |D∗〉〈D∗| (panel a). The second feature is thatthe acceptor is initialized in a state with non-zero excitonic coherence (panel b). AlthoughFig. 3.2 shows results for only a single initial condition, additional simulations (not shown)show that these features hold for arbitrary initial conditions of the donor. In particular,excitonic coherence in the acceptor (and thus coherent beating) is triggered even when theinitial condition in the donor has no such excitonic coherence.

A simple example of the usefulness of the ICC basis is to determine the initial conditionsfor electronic excitation transfer through the FMO complex. The recently discovered 8thchromophore [7, 148] provides a plausible donor to the remainder of the complex since itsits on the side nearest the chlorosome antenna complex [20]. Since structural informationconcerning the location of the FMO complex is limited, standard practice to date has beento choose initial and final conditions for simulation of energy transfer in FMO based onapproximate orientation and proximity of chromophores. The choice of such initial conditionshas varied [2, 20, 96], particularly with regards to whether or not the initial quantum statesinclude any excitonic coherence. Evaluation of the ICC basis between a donor complexconsisting solely of site 8 and an acceptor complex consisting of the remainder of the complex(sites 1-7) implies that the acceptor state is mostly localized on site 1 (see Appendix 3.7.5).We illustrate this in Figure 3.1(a). This initial condition is not an energy eigenstate, so theresulting in vivo dynamics would necessarily start from a state with coherence in the excitonbasis and thus give rise to the quantum beating seen in the laboratory experiments [44].

3.3 Coherent versus thermal transport in a modeldimer with an energy gradient

The way in which this coherence regenerating transfer mechanism can yield a non-trivialbiological role for coherence can be already illustrated with a simple model dimer complexconnected to other complexes. Choosing inter-complex couplings to and from the dimer tobe at individual sites as above implies that initialization and transfer in the ICC basis willbe at these sites. Consider preferred ‘forward’ excitation transfer to be that from the donorat site 2 onward to the next complex. Then the asymptotic probability of successful transfer


through the complex will be proportional to that population. For a dimer, the electronicHamiltonian is given by

H =

[cos θ sin θ− sin θ cos θ

] [0 00 ∆E

] [cos θ − sin θsin θ cos θ

],

where θ is the mixing angle, which measures the intra-dimer delocalization, and ∆E theexciton energy difference. A non-zero mixing angle corresponds to non-zero exciton delocal-ization, as indicated by the inverse participation ratio N = 1/(sin4 θ + cos4 θ).

The dimer admits two extreme models of energy transfer: quantum beating due to coher-ent evolution and instantaneous relaxation to thermal equilibrium between excited electronicstates. Instantaneous relaxation provides an upper bound on the speed of excitation transfergoverned by a classical master equation, since the dynamics governed by such equations aredriven toward thermal equilibrium. This is imposed by the requirement of detailed balancewhich governs classical dynamics, whether between sites as in Förster theory [84] or betweenexciton populations as in variants of Redfield theory [155]. For instantaneous relaxation tothe thermal distribution, the probability that site 2 is occupied is independent of the initialcondition:

pth2 ∝ 〈2|e−βH |2〉 =cos2 θ + eβ∆E sin2 θ

1 + eβ∆E. (3.6)

In contrast, boosts in population due to quantum beating are not restricted by such clas-sical limits [79, 29, 76]. For coherent motion with initialization at site 1, the time-averagedprobability of an excitation at site 2 is

pcoh1→2 = 〈∣∣〈1|e−iHt|2〉∣∣2〉t = 2 cos2 θ sin2 θ, (3.7)

while for initialization at site 2, we have pcoh2→2 = 1 − pcoh1→2. Figure 3.3 plots the differencebetween coherent and thermal population on site 2, as a function of both the intra-dimerdelocalization measure θ and energy difference ∆E, for both possible initial conditions. Itis evident that regardless of initial conditions, for a sufficiently uphill energetic arrangement(∆E > 0) intra-dimer quantum beating will be asymptotically more effective than intra-dimer thermalization in enabling transfer onward from the complex via site 2. The locationof the FMO parameters in Fig. 3.3 shows that the 1-2 dimer of FMO satisfies such anarrangement at 77K and is on the borderline for strictly enhanced transfer due to coherenceat room temperature.

3.4 Design for biomimetic ratchetThe asymmetry between incoherent and coherent population transfer seen above for a simplemodel dimer suggests a design principle that could be exploited for enhanced unidirectionaltransfer [75] and, more generally, a novel type of ratchet based on quantum dynamics [120].


0 Π�16 Π�8 3Π�16 Π�4Mixing angle Θ

-4

-2

0

2

4

Ene

rgy

gap

DE

�k BT

Initial site: 1HaL

áí

à

ì

0 Π�16 Π�8 3Π�16 Π�4Mixing angle Θ

Initial site: 2HbL

áí

à

ì

The

rmal

Coh

eren

t

-1.0

-0.5

0.0

0.5

1.0

Figure 3.3: Difference between coherent and thermal populations on site 2, pcohi→2 − pth2 , asa function of dimer Hamiltonian parameters for (a) initial site i = 1 and (b) initial sitei = 2. The empty symbols � and � indicate location of parameters for the 1-2 dimer in theFMO complex of C. tepidum at 300K as determined in Refs. [2] and [35] respectively. Filledsymbols indicate the corresponding parameters at 77K.

Ratchets and Brownian motors [57] utilize a combination of thermal and unbiased non-equilibrium motion to drive directed transport in the presence of broken symmetry. To takeadvantage of such a ratchet effect, strongly linked chromophores with coherent transfer notlimited by detailed balance should have an uphill energy step relative to the desired directionof transport, whereas weakly linked chromophores with incoherent transfer steps should bearranged downhill.

As a proof of principle, we present an example in which this coherent ratcheting effect re-sults in asymptotic spatial bias of transport. Consider a weakly linked chain of heterodimersbreaking spatial inversion symmetry, as illustrated in Figure 3.4(a). In any classical randomwalk, transition rates must satisfy detailed balance to assure thermal equilibrium. This guar-antees that a classical walk along such a chain is unbiased (Appendix 3.7.3). However, wehave carried out 2CTNL quantum simulations on small chains of dimers that suggest thatincluding the effects of coherence in each dimer breaks the symmetry of detailed balanceto yield a non-zero drift velocity. Since simulations with the 2CTNL approach would becomputationally prohibitive for large numbers of dimers, these simulations were carried outon a chain of three weakly linked dimers with parameters for each dimer matching those ofthe 1-2 dimer of FMO used earlier and an inter-dimer coupling of 15 cm−1. We note thatthis inter-complex coupling strength is well below the cut-off below which energy transportin light harvesting complexes is usually described by completely incoherent hopping withinFörster theory, though without the possibility of coherence regeneration [14, 35]. The resultsof these simulations are used to define left and right inter-dimer transition rates for thecentral dimer. These are then used together with ICC initial conditions from our analysisof the inter-complex coupling as input into a generalized classical random walk describing


àà

à

à

à

à

à

à

à

à

ìììììì

ì

ì

ì

ì

0 500 1000 1500 2000

Time HfsL

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Tra

nsfe

rra

teas

ymm

etry

HbLc = 0c = 0.6c = 0.9

0 100 200 300 400 500 600

Coherence time HfsL0

10

20

30

40

50

Dri

ftve

loci

tyHnm

�nsL

HcL

Figure 3.4: Biased energy transport in an excitonic wire due to spatial propagation of co-herence. (a) A weakly linked chain of heterodimers is arranged such that the higher energystate is always to the right, with a typical inter-dimer distance of 3 nm. The arrow indicatesthe direction of biased transport. (b) Relative asymmetry between left and right inter-dimertransfer rates (Eq. (3.42) of Appendix 3.7.4) as a function of the time before transferring fora dimer excitation initialized in the asymptotic distribution of site populations. (c) Drift ve-locity vs coherence time as modified by bath correlation time (squares) and cross correlationcoefficient between dimer sites (diamonds). The dashed line is a linear fit to guide the eye.Full details of the simulations in panels b and c are in Appendix 3.7.4.

energy transfer along the chain of dimers. The physical model corresponds to the chainshown in panel (a) of Figure 3.4, with red sites (dimer internal site 1) at energy zero, whileblue sites (dimer internal site 2) are at energy 120 cm−1. Full details of the construction oftransition rates and of the set up and solution of the generalized random walk are describedin Appendix 3.7.4.

Formally, this theoretical approach corresponds to using the microscopic quantum dy-namics within and between complexes to define state-specific rates between complexes thatgenerate ‘quantum state controlled’ incoherent energy transfer dynamics over long distances.A feature of this simulation strategy is that we have a priori eliminated the possibility ofreaching true thermal equilibrium, since we do not include the long range coherence termsbetween different dimers. However, it is reasonable to expect the long term influence ofsuch coherences to be negligible, since the inter-dimer couplings are very small. We choosethis hybrid approach to the excited state dynamics since we wish to base our simulationson numerically exact calculations, made here with the 2CTNL method that is valid in bothlimits of strong and weak environmental couplings [76].

These quantum state controlled incoherent dynamics can generate a significant bias in the


spatial distribution of excitation transfer when the intra-dimer dynamics display long lastingquantum coherence. We analyzed the random walk with both Monte Carlo simulations onlong finite chains and an analytic solution [46] of the asymptotic mean and variance ofthe distribution, as detailed in Appendix 3.7.4. Figure 3.4(b) shows that the underlyingasymmetry of transfer rates and violation of detailed balance dynamics is due to the non-equilibrium state of the donor. The bias is in the forward direction, corresponding to theuphill step within dimers. Figure 3.4(c) plots the asymptotic drift velocity of the random walkagainst the timescale of excitonic coherence. We determine this coherence time from a bestfit of the timescale of exponential decay of intra-dimer excitonic coherence. The timescale ofcoherent oscillations was tuned in two ways, (i) by changing the bath correlation time and(ii) by increasing the spatial correlations between the chromophore-bath couplings [74]. Wesee a close correlation between the timescale for quantum beating and the magnitude of thebias, regardless of the underlying physical mechanism used to tune the coherence time. Ingeneral, we cannot rule out the possibility that a non-equilibrium/non-Markovian classicalmodel might also yield such biased transport. (We already ruled out such a possibility for aMarkovian classical model in Appendix 3.7.3.) However, this strong correlation between theduration of quantum beating, independent of its origin, and the asymptotic transport biassupports our interpretation that in this model system the ratchet effect is due to quantumcoherent motion. Indeed, the fact that the drift velocity appears to approach a small or zerovalue as the coherence time goes to zero in Figure 3.4(c) shows that any contribution derivingfrom classical non-equilibrium system/non-Markovian bath dynamics here is extremely smallrelative to that deriving from the quantum coherence maintained in the system degrees offreedom.

Since our results demonstrate a ratchet effect, it is important to consider why this sort ofmotion is not forbidden by the second law of thermodynamics. The answer is that the systemis never allowed to reach thermal equilibrium along the infinite chain of dimers. This featureis shared by the excitations transferred in natural light harvesting systems, which also donot exist for long enough to reach equilibrium. The resulting directed motion shows somesimilarity to the operation of a quantum photocell [136], where coherence can (in principle)allow for enhanced conversion of energy by similarly breaking a limit imposed by detailedbalance. In both cases, no additional source of energy is supplied besides that of the non-equilibrium photon which creates the initial excitation. This contrasts with the operationof typical classical and quantum brownian motors [57], where detailed balance is broken byapplying an additional driving force.

Our results for an infinite chain of heterodimers confirm the effectiveness of our ratchetfor energy transfer, which we ascribe to the combination of intra-complex excitonic coherencewithin dimers and an uphill intra-complex energy gradient. The non-zero drift velocity meansthat over long distances this ratchet offers a quadratic improvement in transfer times overany corresponding classical walk, which is unbiased (Appendix 3.7.3). However, in contrastto the speedup offered by quantum walks [68], this ratchet requires only short ranged andshort lived coherences that will be resilient to the static and dynamic disorder of biologicalenvironments. This spatial bias constitutes a preferential direction for the energy flow across


0 100 200 300 400 500

Time HfsL-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Popu

latio

ns�r

ates

HaL pHD� 2LdpHA� 2L�dt

dpHA� 1L�dt

ÈΨ0\ = È1\0 100 200 300 400 500

Time HfsL0.0

0.2

0.4

0.6

0.8

1.0

Rel

.pop

ulat

ion

on2

HbLÈΨ0\ = È1\ÈΨ0\ = È2\

Figure 3.5: Simulations of FMO dynamics at room temperature. (a) Population of dominantICC donor state at site 2 in the 1-2 dimer (|D2〉〈D2|, solid line) compared with the rate ofpopulation change of its ICC acceptor state in the 3-7 complex (|A2〉〈A2|, dashed line) andthe rate of population change of the other ICC acceptor state not coupled to this donor state(|A1〉〈A1|, dash-dotted line), for the initial condition is |ψ0〉 = |1〉.. The time derivativeshave been scaled to aid comparison of correlations. (b) Population of site 2 relative to thetotal 1-2 dimer population, p2/(p1 + p2), for both choices of initial conditions.

multiple light-harvesting complexes. It could thus be of direct biological relevance for FMO,which serves as a quantum wire connecting the antenna complex to the reaction center. Wetherefore now consider the implications for FMO in more detail.

3.5 Role of coherent energy transport in theFenna-Matthews-Olson complex

We now specifically consider the role of the coherent dynamics in the uphill step energy ofthe FMO complex, which corresponds to the 1-2 dimer in the usual notation (see Fig. 3.1).Since the 1-2 dimer is relatively weakly coupled to the other chromophores in the complex,we may consider transfer to and from this dimer on the basis of our perturbation analysisusing ICC states. By performing a singular value decomposition of the appropriate couplingmatrices (see Appendix 3.7.5), we find that the dominant couplings to and from this dimerare from site 8 to site 1, and from site 2 to site 3. This suggests the relevance of our dimermodel from Section 3.3, where site 8 acts as a donor to the 1-2 dimer, and site 2 in turn actsas a donor to the 3-7 complex. Figure 3.5(a) presents results of a 2CTNL simulation on sites1-7 of FMO, partitioning FMO between donor state on the 1-2 dimer and acceptor states onthe remaining sites 3-7 (i.e., neglecting the prior donation from the 8th site to the 1-2 dimer).The corresponding ICC donor/acceptor states are given in Appendix 3.7.5 (Eqs. (3.47–3.48)).We see that the ICC donor population |D2〉〈D2| ≈ |2〉〈2| is positively correlated with the rateof growth of its coupled ICC acceptor |A2〉〈A2|, but negatively correlated with the growth of


the other ICC acceptor state |A1〉〈A1|, to which it is not coupled. This is in agreement withthe predictions of our theory from Section 3.2. (The small deviations arise because FMO isnot quite in the regime of validity for the perturbation theory and because Eq. (3.5) is notstrictly valid for the situation with two acceptor states |A〉.) The simulation is performed at300K for a bath correlation time of 50 fs as described previously [75].

As indicated by the location of the FMO 1-2 dimer Hamiltonian parameters in Figure 3.3,this particular chromophore dimer appears be optimized to have an uphill energy gradientjust large enough so that excitonic coherence enhances transfer if initialized at site 2 (panel a)without also suppressing transfer initialized at site 1 (panel b). In Figure 3.5(b) we comparethe portion of dimer population on site 2 from 2CTNL calculations with the classical upperbound of the thermal average, for initial conditions in both of the ICC states |1〉 and |2〉. Thepopulations show quantum beating deriving from partially coherent motion. The populationsalso agree with the predications of our simple dimer model (Section 3.3): the populationat site 2 averaged over quantum beats (Eq. (3.7)) is nearly equal to the thermal average(Eq. (3.6)) for initialization at site 1 and greater than the thermal average for initializationat site 2.

As evident from Fig. 3.5, these temporary boosts in population at site 2 due to coherenceshould correspond to enhanced biological function, since they drive excitations preferentiallytoward site 3 instead of backwards toward the antenna complex. The 1-2 dimer in the FMOcomplex thus appears to act as one link of our proposed ratchet for energy transport, therebyenhancing unidirectional energy flow through this pathway of the FMO complex. Consistentwith previous numerical estimates of the contribution of coherent energy transfer to transferefficiency in photosynthetic systems [117, 154], we expect that any quantitative enhancementto the speed of energy transfer through FMO due to such a limited ratchet effect will berelatively small compared to the contribution of incoherent energy relaxation. Refining suchestimates is not the purpose of this work. Rather, our new dynamical model of transportbetween ICC states allows us to propose specific physical advantages that the electroniccoherence provides for general light harvesting systems. In particular, we established theability to propagate excitonic coherence between weakly coupled sub-units and to use theratchet effect to enhance unidirectional transport. It is also conceivable that the cumulativecontribution of many such small contributions from propagating coherence through the entirephotosynthetic apparatus of green sulphur bacteria (of which FMO is only a small part)could indeed make a major contribution to the speed of energy transfer, as in the full ratchetexample.

3.6 ConclusionsWe have proposed a microscopic mechanism for the propagation of excitonic coherence inenergy transfer between photosynthetic complexes. The mechanism allows coherence to bepropagated between sub-units of a large light harvesting “supercomplex” that is composedof multiple complexes that individually support coherence. Our analysis shows that the


key role in the inter-complex transfer is played by the inter-complex coupling (ICC) basis,rather than energy or site bases employed by prior analyses. By utilizing ICC donor andacceptor states, we showed that coherence can enable biased energy flow through a ratchetmechanism. We provided evidence that this same principle acts to ensure unidirectionalenergy flow in the FMO complex. Since one-way transmission of electronic energy from theantenna complex to the reaction center constitutes the main function of the FMO proteinin the light harvesting apparatus of green sulfur bacteria, this supports a biological role forthe electronic quantum coherence in this particular light harvesting system.

Propagating coherence provides both a mechanism by which coherent motion can influ-ence transfer rates and photosynthetic efficiency in light harvesting systems of arbitrary size(a possible quantum advantage), and a scalable method for multi-scale modeling of suchexcitonic systems without neglecting the contributions of coherence (practical simulations).Our example and analysis of a coherently enabled ratchet effect along a chain of heterodimersdemonstrates both of these features. This proof of principle model shows that even short-lived excitonic coherence can, since it propagates spatially, lead to large scale dynamics thatare incompatible with any completely classical description. We also demonstrated how fullyquantum models need only to be used for tightly coupled sub-complexes, while transfer be-tween sub-complexes may take the form of classical hops with connections between states ofdifferent sub-complexes constrained by the ICC basis.

Similar techniques should allow us to assess the role of coherence in natural photosyn-thetic super-complexes with hundreds of chlorophyll molecules, such as arrays of LH1 andLH2 rings in purple bacteria, and the photosystem I and II super-complexes of higher plants[9]. For example, direct calculation of ICC states should help us evaluate the significanceof long lasting coherences in bacterial reaction centers [86, 87], since these systems are alsousually unlikely to absorb light directly [9] and thus might be benefitting from recurrenceof coherence propagated from light harvesting complexes. Some bacterial reaction centersalso feature an uphill step opposing the direction of desired energy flow [147], resembling theenergetic arrangement in the uphill step of the FMO complex.

Finally, the dynamics in our chain of heterodimers model constitute a novel type ofratchet that utilizes spatial propagation of quantum coherence in place of a driving forceand as such are also of more general interest. Thus, in addition to excitonic systems, weexpect that a similar ratchet effect could be demonstrated in other experimental systemsdescribed by spin-boson Hamiltonians, such as cold atoms in optical lattices [137, 119].

3.7 Appendices

3.7.1 Extending multichromophoric Förster theory

In this Appendix, we provide additional technical details of the derivation of Eqs. (3.1–3.2).Consider a system under zeroth order Hamiltonian H0 with perturbation Hamiltonian V . In


the interaction picture ρI(t) = eiH0t/~ρ(t)e−iH0t/~ the von Neumann equation is

dρIdt

= − i~

[VI(t), ρI(t)], (3.8)

which can be formally integrated to yield,

ρI(t) = ρI(0)− i

~

∫ t

0

dt′[VI(t′), ρI(t

′)]. (3.9)

Inserting Eq. (3.9) into Eq. (3.8), keeping all terms second order in V and transforming backto the Schröedinger picture yields the second order contribution to the time-convolutionlessequation of motion

dρ

dt= − 1

~2

∫ t

0

dτ [V, [e−iH0τ/~V eiH0τ/~, ρ(t)]], (3.10)

where we approximated e−iH0τ/~ρ(t− τ)eiH0τ/~ ≈ ρ(t) (valid to this order in V ).We are interested in the lowest order contribution to the donor and acceptor electronic

states from a perturbative treatment of the inter-complex coupling terms with the zerothorder Hamiltonian given by that of the otherwise independent donor and acceptor complexes.Since a first order treatment of Hc gives coherences between the donor and acceptor butno population transfer, we thus consider the second order contribution. Note that ourperturbation parameter is this inter-complex coupling Hc rather than the usual couplingto the bath, so our results will apply to any bath coupling strength. Substituting ourperturbation V = Hc into Eq. (3.10) and tracing over the bath yields the equation of motionfor acceptor states,

dσkk′

dt=∑j

∑j′′k′′

Jj′′k′′

~2

∫ t

0

dτ TrB

[Jjk〈Dj|ρeDρ

gAe−iH0τ/~|Dj′′〉〈Ak′′ |eiH0τ/~|Ak′〉

+ 〈Ak|e−iH0τ/~|Ak′′〉〈Dj′′ |eiH0τ/~ρeDρgA|Dj〉Jjk′

], (3.11)

where we used the initial condition ρ = ρeDρgA for a general excited donor state with the

acceptor in the ground state at thermal equilibrium [78]. The donor equation is similar andthus omitted for conciseness. To simplify these equations, we use the following identity,which follows from the substitution H0 = HA + HD, by employing the cyclic properties ofthe trace as well as the assumptions that the donor and acceptor baths are independent andthat all strictly donor and strictly acceptor terms commute,

TrB

[〈Dj|ρeDρ

gAe−iH0τ/~|Dj′′〉〈Ak′′ |eiH0τ/~|Ak′〉

]= TrB

[eiH

gDτ/~〈Dj|ρeDe−iHDτ/~|Dj′′〉

]TrB

[e−iH

gAτ/~〈Ak′′ |eiHAτ/~|Ak′〉ρgA

]. (3.12)


Substitution of this identity and its Hermitian conjugate into Eq. (3.11) gives a form amenableto substitution by products of acceptor and donor lineshape functions [78], given by

Ik′k

A (ω) =

∫ ∞−∞

dt eiωt TrB

{eiH

gAt/~〈Ak′|e−iHAt/~|Ak〉ρgA

}(3.13)

Ej′jD (t, ω) =2

∫ t

0

dt′e−iωt′TrB

{e−iH

gDt/~〈Dj′|eiHDt/~ρeD|Dj〉

}. (3.14)

Inserting these lineshapes yields Eqs. (3.1–3.2). Explicit dependence upon t in the donorlineshape can be removed by applying the Markov approximation, that is, allowing the upperlimit of this integral to be extended to infinity and assuming that the donor ρeD is stationary.This would give rate expressions corresponding to those of equilibrium multichromophoricFörster theory [78].

We note that the result in Eqs. (3.1–3.2) shows that these equations do not necessarilyconserve positivity, a feature hidden by the sum over states to determine an overall trans-fer rate [78]. This is an intrinsic limitation of the perturbative approach to inter-complextransfer. In particular, these expressions may predict the creation of non-physical acceptorcoherences of the form |Ak〉〈Ak′| even without necessarily increasing both of the correspond-ing population terms |Ak〉〈Ak| and |Ak′〉〈Ak′|. For this reason, in determining ICC states,we explicitly only consider those states which will experience population growth or decay.

3.7.2 Weak system-bath coupling

Under an approximation of weak system-bath coupling relative to the electronic Hamiltonianof the isolated donor He

D, the full density matrix can be factorized in the form ρ(t) = ρBeqσ(t)between the equilibrium state of the bath ρBeq = ρgDρ

gA and the electronic state of the system

σ(t). We do not need to assume weak system-bath coupling for the acceptor, since it isalready in factorized form in the electronic ground state. Likewise, we do not need to assumeweak system-bath coupling relative to the inter-complex coupling Hc, since Hc does notenter into lineshape expressions for either the donor or the acceptor. Accordingly, Eq. (3.10)becomes,

dσ

dt= − 1

~2[V, [K(t), σ]], (3.15)

K(t) =

∫ t

0

dτ TrB[e−iH0τ/~V eiH0τ/~ρBeq], (3.16)

where the explicit perturbation V is still the inter-complex coupling Hc. The Markov ap-proximation is given by taking t → ∞, in which case we write K = limt→∞K(t). Since Vand K are not in general equal, the Markovian expression is not in Lindblad form and thusdoes not necessarily conserve positivity, a standard limitation of perturbative derivations ofquantum master equations [12].


For convenience, from now on we apply the Markov approximation. Similar conclusionshold in the more general case. We then can write Eq. (3.15) in terms of the evolution ofeach density matrix element as

dσabdt

=1

~2

∑cd

Rabcdσcd (3.17)

by defining Redfield-like tensor elements

Rabcd = −∑e

[δdbVaeKec + δacKdeVeb] +KacVdb + VacKdb. (3.18)

To evaluate our particular model of inter-complex transfer, it is useful to define an acceptorlineshape that only depends upon the bath state in the same form as the donor lineshape(Eq. (3.14)),

E j′jD (ω) =

∫ ∞−∞

dt eiωt TrB

{eiH

gDt/~〈Dj′ |e−iHDt/~|Dj〉ρgD

}. (3.19)

For weak system-bath coupling and in this Markov limit, we can write the general donorlineshape ED(t, ω) (Eq. (3.14)) in terms of this modified lineshape,

Ej′jD (∞, ω) =

∑j′′

E j′jD (ω)σjj′′ . (3.20)

Since K is Hermitian, to evaluate the model of Section 3.2 it suffices to calculate Kjk =〈Dj|K|Ak〉. Using the cyclic property of the trace and inserting the donor and acceptorlineshapes, we find

Kjk =1

4π

∑j′k′

Jj′k′

∫ ∞−∞

dω E j′jD (ω)Ik

′kA (ω). (3.21)

We can now evaluate the tensor elements in Eq. (3.18) that specify the influence of donordensity matrix elements on inter-complex transfer, either by using the matrix elements forK given in Eq. (3.21) or by using the equivalence in Eq. (3.20) to insert the modified donorlineshape into Eqs. (3.1–3.2). The relevant tensor elements for the change of the acceptorelements due to the donor are given by,

Rkk′j0j′0=∑k′′j

Jjk′′

4π

∫ ∞−∞

dω[Jj′0k′ E

j0jD (ω) Ikk

′′

A (ω) + Jj0k Ejj′0D (ω) Ik

′′k′

A (ω)]

(3.22)

and for the change of the donor itself due to donating an excitation,

Rjj′j0j′0= −

∑kk′j′′

Jj′′k′

4π

∫ ∞−∞

dω[δj′j′0Jjk E

j0j′′

D (ω)Ikk′

A (ω) + δjj0Jj′k Ej′′j′0D (ω)Ik

′kA (ω)

]. (3.23)


Since these tensor elements are given in terms of an arbitrary basis for the donor and acceptorelectronic states, we may write them in terms of the ICC states for which J is diagonal.Considering the elements that affect populations (k = k′ for acceptor, j = j′ for donor), it isthen evident that in the ICC representation, the factors Jj0k restrict nonzero contributionsfrom donor states j to only those deriving only from the coupled donor ICC state |Dj0〉〈Dj0|.Each of these terms also includes a sum over other inter-complex coupling matrix elementsand lineshapes, but these only affect the magnitude of the allowed transitions. In the casewhere there is only a single nonzero ICC coupling, Eq. (3.17) thus reduces to Eq. (3.5) ofthe main text.

Note that since our donor and acceptor lineshapes (Eq. (3.19) and Eq. (3.13)) takeidentical forms under weak environmental coupling, resulting forward and backward transferrates will be equal and thus may not necessarily respect detailed balance. Therefore we donot calculate actual rates using Eq. (3.19).

3.7.3 Proof that classical Markovian transport is unbiased

Consider a classical Markov process that models transport along a chain of dimers like thechain we used for the quantum coherent ratchet model (Figure 3.4). We impose only onerequirement on the transition rates in this model: thermal equilibrium must be a steady state.For a Markov process, this implies that the transition rates satisfy detailed balance. On eachdimer, we consider two states in the single excitation subspace, which could equally well besites or excitons. For simplicity, consider excitation transfer only between nearest-neighborstates (similar symmetry constraints guarantee unbiased transport even in the general case).Then an excitation initially at the lower energy state of each dimer has two possible moves:with probability p to the higher energy state of the same dimer, or with probability 1− p tothe higher energy site of the neighboring dimer to the left (the non-zero coupling guaranteesthat eventually the excitation will move). For an excitation at the higher energy state of adimer, detailed balance requires that the rate of transitions to each neighboring state (at thelower energy) is the rate from those states scaled by the Boltzmann factor eβ∆E. The relativeintra- vs inter-dimer transition rates are still the same, so the probability of an intra-dimerjump is still p, and 1 − p for the inter-dimer jump, now to the right. Since every jumpalternates between high and low energy states, and these intra- and inter-dimer transitionsare alternatively to the left and to the right, on average the random walk must be stationary.

3.7.4 Ratchet methods

Propagation of coherence With weak coupling between different dimers, inter-dimertransfer should follow the principles of our theory of propagated coherence described inSection 3.2. Here we restrict the inter-dimer coupling to be between nearest neighbors, tosimplify the singular value decomposition. This is a reasonable approximation for realisticdipole-dipole couplings in light harvesting arrays because the 1/r3 scaling ensures a rapidfall-off with inter-chromophore distance. Accordingly, an ICC analysis tells us that after


an inter-dimer transfer the dynamics will be reset with the initial condition on the site inthe dimer nearest the side from which the excitation was received. Thus if an excitationis received from the dimer to the left (right), we restart dynamics the initial condition ison the left (right) site of the new dimer. For the complex consisting of the nth dimer, thiscorresponds to the explicit inter-complex coupling matrix (in the ICC basis)

J =∑

ε∈{−1,+1}

J |n, ε〉〈n− ε,−ε|, (3.24)

where J is an arbitrary inter-dimer coupling strength and |n, ε〉 is the state correspondingto occupation of the right (ε = −1) or left (ε = +1) site of dimer n. We also assumethat upon excitation at a site in a new dimer, the baths of the donating chromophore willinstantaneously revert to thermal equilibrium. Accordingly, since the chain of dimers isperiodic, we can build overall dynamics in this manner from full quantum calculation of justfour time-dependent transfer rates corresponding to left or right transfer to a neighboringdimer from each of the two initial conditions on sites.

Scaled 2CTNL calculations For computational feasibility, we based our calculationsof transfer rates on scaling the results of 2CTNL simulations on a three dimer (six site)system. We may denote the left, central and right dimers as −1, 0,+1, respectively. Weuse the 2CTNL method because it accurately models dynamics under both strong and weakenvironmental coupling [76]. Since we need to calculate rates neglecting back-transfer whilethese simulation methods describe full system dynamics, we calculate the dynamics herewith the inter-dimer coupling J0 set to be very small so that back-transfer was negligible.To begin, we need cumulative transition probabilities F 0

εδ(t) for the transition from initialconditions ε ∈ {+1,−1} to neighboring dimer δ ∈ {+1,−1} at time t. From our simulationson the three dimer chain, the quantity F 0

εδ(t) is simply the total population at time t on dimerδ obtained by starting with initial condition on site ε of the central dimer. The probabilitydensity as a function of t, which is the transition rate, is then given by f 0

εδ(t) = ∂∂tF 0εδ(t)

and evaluated numerically. We then scale the transfer rate to obtain the rescaled ratefεδ(t) = (J/J0)2f 0

εδ(t) corresponding to the coupling J . The rescaled cumulative transitionprobability is obtained by numerical integration, Fεδ(t) =

∫ t0dτfεδ(τ). This scaling of the

transfer rate assumes that to lowest order in J the transfer rate is proportional to J2, asgiven by Eq. (3.1). For our parameters, we found that the scaled transfer rate fεδ(t) doesindeed converge as J0 → 0 and that using a value J0 = 1 cm−1 was sufficiently small to makeany error negligible. This method neglects the effects of excitation loss on the dynamicsof the donating dimer, which is reasonable to lowest order in J . We simulated the threedimer chain using two such 2CTNL calculations, one for each initial condition on a specificsite of the central dimer. Calculations including spatially correlated baths on each dimerwere performed as described previously [74]. In principle, one could perform calculationstaking into account static disorder, but we do not expect static disorder would influenceour qualitative findings since the primary effect of disorder is to limit delocalization and ourscaling procedure already constrains exciton delocalization to individual dimers.


0 200 400 600 800Time HfsL

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Transferratef edHtLH

ps-1 L

HaL e=+1, d=+1e=+1, d=-1

0 200 400 600 800Time HfsL

HbL e=-1, d=+1e=-1, d=-1

Figure 3.6: Inter-complex transfer rates fεδ(t) with initial conditions (a) ε = +1 and (b)ε = −1 for steps δ = +1 (solid lines) and δ = −1 (dashed lines). These rates are derivedfrom 2CTNL simulations for a chain of three dimers with correlation time 50 fs and no spatialcorrelations, as described in the text. The transfer rates oscillate, corresponding to quantumbeatings in the donor, but eventually converge to the same equilibrium rate, as required tosatisfy detailed balance. However, at early times, the left (δ = +1) and right (δ = −1)transfer rates are not equal, oscillating out of phase. When averaged over the limitingdistribution πε of the initial condition this gives rise to the marked short time asymmetry inthe left and right inter-dimer transfer rates that is shown in Figure 3.4(b) of the main text.This asymmetry, although small in absolute terms, is amplified by being repeated over manyhops between dimers and is responsible for the asymptotic bias of the random walk.

Figure 3.6 gives an example of these transfer rates fεδ(t). The transfer rates oscillate,corresponding to quantum beatings in the donor, but eventually converge to the same equilib-rium rate, as required to satisfy detailed balance. However, at early times, the left (δ = +1)and right (δ = −1) transfer rates are not equal, oscillating out of phase. When averagedover the limiting distribution πε of the initial condition this gives rise to the marked shorttime asymmetry in the left and right inter-dimer transfer rates that is shown in Figure 3.4(b)of the main text. This asymmetry, although small in absolute terms, is amplified by beingrepeated over many hops between dimers and is responsible for the asymptotic bias of therandom walk.

Generalized random walk With each transition to a neighboring dimer only dependingon the initial conditions at each dimer, the dynamics constitute a type of Markov chaincontrolled random walk known as a semi-Markov process [134]. In each step of the randomwalk, we start with a state of the form |n, ε〉 denoting occupation of the right (ε = −1)or left (ε = +1) site of the nth dimer. The cumulative transition probabilities Fδε(t) fortransitioning from |n, ε〉 to the left or right dimer ICC acceptor state |n + δ, δ〉 for δ = ±1(see Eq. (3.24)) are determined by the rescaled 2CTNL calculations as described above. The


update ε′ = δ is the initial condition for the excitation on the new dimer following from ourICC analysis. The walk is memory-less in terms of a two-dimensional clock variable (n, t)denoting “space-time” position, but the initial condition at each dimer nevertheless functionsas an additional “coin” degree of freedom ε that controls the likelihood of jumps to a newclock state (n′, t′).

Monte-Carlo algorithm Given the transition probabilities Fεδ(t) for this random walk,we used two techniques to calculate the long time behavior of the overall random walk. First,we performed Monte-Carlo simulations of the evolution for a total time T by averaging overtrajectories of many jumps. We start by setting the clock to the state (n, t) = (0, 0) and thecoin to ε = +1. We sample from the distribution of possible space-time shifts ξεδ = (δ, tεδ)by choosing a pair (u1, u2) of independent uniformly distributed random numbers between0 and 1. For convenience, we define the final transition probability pεδ ≡ limt→∞ Fεδ(t). Ifu1 ≤ pε,+1, we choose δ = +1 for this jump; otherwise, δ = −1. The time tεδ it takes forthis jump is determined by numerically solving the equation u2 = Fεδ(tεδ)/pεδ for t. Wethen update the clock (n′, t′) = (n+ δ, t+ tεδ) and the coin ε′ = δ. This process is repeateduntil time t+ tεδ > T , at which point we record the location of the previous dimer n as thefinal state of that trajectory. The probability density of the final distribution over dimersis derived by binning over many such trajectories (∼5000). Empirically, our Monte-Carlosimulations suggest that the distribution converges to a normal distribution characterizedby its mean and variance, as expected from a central limit theorem for weakly dependentvariables [42].

Analytic model Second, we calculated the mean and variance of final distribution ana-lytically in the asymptotic limit of the total walk time T →∞, using the method suggestedin Ref. [46]. Since the results of these calculations agreed with the Monte-Carlo simulationsbut were much faster, we use this second method for the plots in Figure 3.4. To begin, wecalculate the moments of the transition time tεδ for each jump ε→ δ,

E(tεδ) =1

pεδ

∫ ∞0

tfεδ(t)dt (3.25)

E(t2εδ) =1

pεδ

∫ ∞0

t2fεδ(t)dt (3.26)

by numerical integration. Now, note that transitions between coin states can be describedas a Markov chain with transition matrix P with entries given by the final transition prob-abilities pεδ = limt→∞ Fεδ(t) as defined above, i.e.,

P =

(p+1,+1 p+1,−1

p−1,+1 p−1,−1

). (3.27)

Accordingly, the limiting distribution πε over the coin space is given by the left eigenvectorof P with eigenvalue 1, that is, the solution π of the equation πε′ =

∑ε πεpεε′ . The quantity


πεpεδ gives the probability of the step ε → δ in the limiting distribution. Recalling thedefinition of the space-time shift ξεδ = (δ, tεδ) associated with the step ε → δ, we obtain anaverage space-time shift ξεδ = (δ,E(tεδ)) for this step. Since the coin will converge to thelimiting distribution πε, we now obtain the average space-time shift over all steps as

ξ = E(ξ) =∑εδ

πεpεδ ξεδ ≡ (n, t). (3.28)

Now let nT denote the spatial position of the random walk after a total time T . This randomwalk is the sum of T/t independent steps on average, each of which has an average spatialshift n. Since the expectation adds linearly, we then obtain the average position of the overallwalk as

E(nT ) = nT/t. (3.29)

Figure 3.4 plots the corresponding drift velocity v = E(nT )/T .Given that our random walk appears to converge to a normal distribution, we can fully

characterize the distribution with its mean, calculated above, and its variance. As a practicalmatter, the variance indicates the width of the distribution and thus determines whether ornot a non-zero drift velocity would be observable experimentally. To calculate the variance,we consider two sources of space-time deviations,

ηεδ = ξεδ − ξεδ (3.30)µεδ = ξεδ − ξ, (3.31)

corresponding to deviations ηεδ of the space-time shift of a particular transition from itsaverage value, and deviations µεδ of the average space-time shift for a particular transitionfrom the average space-time shift over all transitions. Since successive steps are weaklycorrelated by the ICC conditions, the latter quantity must be averaged over all possiblesteps in all possible trajectories. We therefore define µ as the single step average obtainedby summing µεδ over all possible sequential steps:

µ = limn→∞

1

n

n∑i=1

µεiδi . (3.32)

Note that for a standard Markov chain with no correlation between steps, this single stepaverage reduces to µ. Assuming the covariation between ηεδ and µ is small, we can combinethem to calculate the overall space-time covariance matrix,

Var(ξ) = Var(ξ − ξ) (3.33)= Var(η + µ) (3.34)= Var(η) + Var(µ). (3.35)


Averaging over the limiting distribution πε, we find for the first contribution to the variance,

Var(η) =∑εδ

πεpεδ Var(ηεδ) (3.36)

where

Var(ηεδ) =

(0 00 Var(tεδ)

), (3.37)

with Var(tεδ) = E(t2εδ) − E(tεδ)2. Now consider the second contribution to the variance,

Var(µ). The variance of the single step average µ, Eq. (3.31), introduces a double sumover products of deviations µεδ. With a correlated random walk, the products of deviationsat different space-time values are also correlated and hence evaluation of these requiresenumeration of all possible jumps connecting them, where these are determined by thetransition probability matrix P , Eq. (3.27). This enumeration, which constitutes a multi-state generalization of the variance for weakly dependent processes [42], is given explicitlyby

Var(µ) =∑εδ

πεpεδµTεδµεδ +

∑εδρσ

∑m≥0

πεpεδp(m)δρ pρσ[µT

εδµρσ + µTρσµεδ], (3.38)

where p(m)δρ = (Pm)δρ and we sum each variable ε, δ, ρ, σ over ±1. The second term in Eq. 3.38

sums up all contributions that m-steps apart, where these are specified by the Chapman-Kolmogorov equation [12]. We note that for convenience, instead of explicitly performing thesum over m, one can equivalently replace the term

∑m p

(m)δρ in the equation above with Qδρ

[46], where Q = (1−P ∗)−1 and P ∗ is the non-equilibrium portion of P , that is, with entriesp∗εδ = pεδ − πε. Combining Eqs. (3.36) and (3.38) into (3.36) yields a space-time covariancematrix Var(ξ) for the two dimensional shift variable ξ. This covariance matrix has explicitentries,

Var(ξ) =

(Var(n) Cov(n, t)

Cov(n, t) Var(t)

). (3.39)

To calculate the final spatial variance Var(nT ), we must take into account the uncertaintyassociated with the number m of discrete hops that happened in time T , in addition to theuncertainty over n. To correctly incorporate both contributions, we calculate the variance ofthe spatial displacement n over a single coin shift over the full two-dimensional coin space,

Var(n,t)(n) = Var(n,t)(n− nt/t)= Var(n)− 2(n/t) Cov(n, t) + (n/t)2 Var(t)

= (1,−n/t) Var(ξ)(1,−n/t)T, (3.40)

where in the first step we subtracted the average value of n over the coin space. We write thevariances over the full coin space (n, t) to emphasize that they are distinct from terms like


Var(n), which is only over the spatial degree of freedom n. Since the variance adds linearlyover m ≈ T/t independent steps, we obtain the variance of the distribution after time T as

Var(nT ) = (1,−n/t) Var(ξ)(1,−n/t)TT

t. (3.41)

The diffusion coefficient for the walk is then given as D = Var(nT )/2T .Figure 3.7 plots the full results of scans over correlation time and cross-correlation coef-

ficients used to create Figure 3.4 of the main text. We see that the width of the excitationtransfer distribution is approximately constant over all parameter choices at about 60 nmafter 1 ns, and that the asymmetry between initial conditions ∆π = π+1 − π−1 accounts formost of the variation in drift velocity. Figure 3.4(b) of the main text is a plot of the relativetransfer rate asymmetry for the limiting distribution of the initial condition πε,

A(t) =

∑εδ πεδfεδ(t)∑εδ πεfεδ(t)

, (3.42)

where the sums are over ε, δ ∈ {−1,+1} as usual.

3.7.5 FMO Hamiltonian and singular value decompositionsThe FMO complex exists in a trimer arrangement, where each monomer contains 7 bac-teriochlorophyll molecules, and three additional BChl molecules (termed the 8th Bchl foreach of the three monomers) are each located between a distinct pair of monomers [148]. Inthis paper, we use a Hamiltonian for a monomer of the FMO complex of C. tepidum calcu-lated by Adolphs and Renger [2], augmented by dipole-dipole couplings to the 8th BChl sitecalculated using their same methodology with the crystal structure of Tronrud et al. [148].We assign each of the three BChl 8 pigments to the monomers with which they have thestrongest dipole-dipole coupling. The Hamiltonian matrix is given below in units of cm−1

above 12 210 cm−1, where elements of the matrix are indexed according to site from 1 to 8:

200 −87.7 5.5 −5.9 6.7 −13.7 −9.9 37.5−87.7 320 30.8 8.2 0.7 11.8 4.3 6.5

5.5 30.8 0 −53.5 −2.2 −9.6 6. 1.3−5.9 8.2 −53.5 110 −70.7 −17. −63.3 −1.86.7 0.7 −2.2 −70.7 270 81.1 −1.3 4.3−13.7 11.8 −9.6 −17. 81.1 420 39.7 −9.5−9.9 4.3 6. −63.3 −1.3 39.7 230 −11.337.5 6.5 1.3 −1.8 4.3 −9.5 −11.3 ?

. (3.43)

The energy of site 8 is marked with a question mark to indicate that it is unknown, sinceit has not been calculated. Accordingly, our simulations of the full FMO complex use onlythe portion of this Hamiltonian for sites 1-7, as in previous studies [75].

To determine donor and acceptor ICC states for a given coupling matrix J , we performthe singular value decomposition J = UDJU

†A =

∑l Jl|Dl〉〈Al| as described in Section 3.2.


Here are the results of two examples we use with our FMO Hamiltonian. Let the notationJAD denote the coupling matrix from the donor (D) rows and the acceptor (A) columns ofEq. (3.43). As plotted in Figure 3.1, for the coupling from site 8 to sites 1-7, we haveJ1-7

8 = J∗|D∗〉〈A∗| with

J∗ = 41.9 |D∗〉 = |8〉 |A∗〉 =

−0.912−0.158−0.0310.043−0.1050.2290.275

, (3.44)

with entries 〈i|A∗〉 for states |i〉 = |1〉, . . . , |7〉. With only a single donor site, the ICCacceptor (donor) state from the singular value decomposition is as simple as the normalizedvector corresponding to the dipole-dipole matrix. Since occupation probabilities correspondto these amplitudes squared, the acceptor among sites 1-7 is mostly (83%) on site 1, as shownin Fig. 3.1(a).

A less trivial example is given by considering the 1-2 dimer as a donor to and acceptorfrom the remainder of the complex 3-8. In this case, we have J3-8

1-2 =∑

i=1,2 Ji|Di〉〈Ai| with

J1 = 43.6 |D1〉 =

[−0.969−0.247

]|A1〉 =

−0.2970.085−0.1530.2380.196−0.887

, (3.45)

J2 = 34.3 |D2〉 =

[0.247−0.969

]|A2〉 =

−0.832−0.2740.028−0.432−0.1930.089

, (3.46)

with entries corresponding to states |i〉 in ascending order. Thus the coupling in the ICCbasis is mostly from site 8 (78%) to site 1 (94%), and from site 2 (94%) to site 3 (69%). If


we omit site 8 from the acceptor, these acceptor and donor states are modified as follows:

J1 = 19.7 |D1〉 =

[0.9950.099

]|A1〉 =

0.433−0.2570.342−0.633−0.479

, (3.47)

J2 = 34.4 |D2〉 =

[0.099−0.995

]|A2〉 =

−0.876−0.254−0.001−0.381−0.153

. (3.48)

We use these ICC states in Fig. 3.5 since only sites 1-7 are included in the 2CTNL simulation,as the site energy of the 8th BChl is unknown (see Eq. (3.43)), and as it is furthermore unclearwhether this 8th BChl is present in all cases in the natural system [148].


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Figure 3.7: Full results of simulations used for analysis of the unidirectional random walk,as used to create Figure 3.4 of the main text. The parameters of each dimer match that ofsites 1-2 in FMO, as described in the main text. We used a Debye spectral density with areorganization energy of 35 cm−1 and variable bath and spatial correlations, as indicated onthe figure. For the left panels we have variable time correlation and no spatial correlations.For the right panels we vary the spatial correlation (see Ref. [74]) and fix the correlationtime at 50 fs. (a,b) Drift velocity, from Eq. (3.29). (c,d) Standard deviation σ =

√Var(nT )

of the walk at 1 ns, from Eq. (3.41). (e,f) Coherence time τ , from a least squares fit of theexponential decay of excitonic coherence, |ρe1e2| ∼ Ae−t/τ + B: we evaluate this here andin Figure 3.4 for t > 100 fs to exclude non-exponential decay. (g,h) Asymptotic transferasymmetry ∆π = π+1 − π−1 indicating the overall preference for right over left transfer.

54

Part II

Probing and controlling quantumcoherence

55

Chapter 4

Inverting pump-probe spectroscopy forstate tomography of excitonic systems

SummaryWe propose a two-step protocol for inverting ultrafast spectroscopy experiments on a molec-ular aggregate to extract the time-evolution of the excited state density matrix. The firststep is a deconvolution of the experimental signal to determine a pump-dependent responsefunction. The second step inverts this response function to obtain the quantum state ofthe system, given a model for how the system evolves following the probe interaction. Wedemonstrate this inversion analytically and numerically for a dimer model system, and evalu-ate the feasibility of scaling it to larger molecular aggregates such as photosynthetic protein-pigment complexes. Our scheme provides a direct alternative to the approach of determiningall Hamiltonian parameters and then simulating excited state dynamics.

4.1 IntroductionUltrafast nonlinear spectroscopy allows us to experimentally observe excited state dynamicsin molecular aggregates, and in particular, energy transfer essential to the function of naturallight harvesting systems [98, 9, 93]. The existence of these experimental tools prompts a nat-ural question: is it possible to use spectroscopic measurements to directly infer the excitedstate of such systems? A complete answer to this question would be a procedure for quan-tum state tomography (QST), that is, for reconstruction of the full density matrix describingthe quantum state [103, 145]. State tomography is a technique that has found widespreadapplication for validating and characterizing quantum devices designed as components forquantum computation. Such full characterization of an exciton state over multiple pigments,beyond a mere classical probability distribution, would offer information essential to under-standing the explicitly quantum features of energy transport, which include coherence [44],entanglement [125] and possibly other types of non-trivial quantum dynamics [68, 67, 11,

CHAPTER 4. INVERTING PUMP-PROBE SPECTROSCOPY 56

101]. In this work, we show that under appropriate conditions and assumptions, QST ofexcited states can be performed from the results of a series of pump-probe type ultrafastspectroscopies.

The most sophisticated non-linear technique for resolving energy transfer dynamics is thetwo-dimensional (2D) photon-echo, in which the time delays between three ultrafast pulsesare manipulated to provide a 2D map between pump and probe frequencies at fixed timedelays [30, 1]. These two-dimensional maps allow for direct visualization of the relationshipbetween excitation and emission energies as a function of delay time. More formally, 2Dspectroscopy is usually interpreted in the limit of impulsive interactions. In this approxima-tion, it provides snapshots of the 3rd-order response function [98]. Important applicationsof 2D spectroscopy to photosynthetic systems have included resolving energy transfer path-ways [14] and the dynamics of electronic quantum beats [44, 129, 107, 37]. In contrast,pump-probe spectroscopy (also known as transient absorption) is a simpler type of 3rd-orderspectroscopy that historically predates 2D. In a pump-probe setup, a pump pulse excites thesystem, which is probed at some time later by a probe pulse. Because of its relative easeof experimental implementation, pump-probe was used to follow ultrafast energy transferdynamics in photosynthesis long before 2D spectroscopy. For example, it provided the firstevidence of electronic quantum beats in photosynthetic pigment-protein complexes, in 1997[126]. Pump-probe provides less information than the 2D photon echo, because the pump-probe signal can be obtained by integrating over the excitation axis in a 2D spectra [130].However, for the purposes of this work, pump-probe has a clear advantage, namely, that itcan be directly interpreted as a measurement of the state created by the pump pulse. For-mally, the pump dependence is entirely contained within the change in the density matrixof the system after interacting with the pump [80].

In the past, time-resolved spectra such as pump-probe have been analyzed by simultane-ously or concurrently fitting spectral components, known as decay associated or species asso-ciated spectra, with a kinetic model [143, 153]. These techniques are powerful, as evidencedby their widespread application to experiments. However, much of the kinetic informationthey reveal can be seen more directly in 2D spectra. Moreover, kinetic models, although ad-equate for many purposes, cannot describe more complex dynamics, such as those derivingfrom quantum beats or from a non-Markovian bath. Our approach to QST side-steps theissues of extending such integrated analyses by focusing on identifying the quantum statedirectly.

Recently, it was shown that a combination of photon-echo measurements of excitonic sys-tems can be combined to perform quantum process tomography of excitonic dimers, eitherby using differently colored pulses [157] or by combining peak amplitudes from a set of 2Dspectra [156]. Process tomography [103, 95] is more general than state tomography, becauseit specifies the full set of possible quantum evolutions for a system given any initial condition.This makes it well suited to characterizing gates for quantum computation, which are bydefinition designed to handle any possible input state. However, determining the full processmatrix is expensive: it requires at least d4 − d2 real parameters for a d-dimensional Hilbertspace, in contrast to d2 parameters for state tomography. Moreover, for analysis of complex


molecular dynamics in condensed phases, such information, although potentially helpful, isnot necessary, because most trajectories contained in the process matrix start from initialconditions that are implausible for a molecular aggregate. Indeed, typical theoretical inves-tigations of dynamics in light harvesting systems [75, 142] follow dynamics after excitationfor only a limited set of plausible initial states, such as the states which absorb sunlightor excitations from neighboring antenna complexes. Finally, the relative simplicity of statetomography helps to simplify consideration of new theoretical approaches to tomography,particularly because process tomography is often based on a series of state tomographies[95].

In this paper, we present a new approach to state tomography of excitonic systems basedon pump-probe spectroscopy. Our approach is based on a two stage protocol that separatesthe easy (field based) and hard (system based) parts of the inversion process. This yieldsseveral advantages over prior approaches [157, 156], including the ability to use arbitrarilyshaped laser pulses and to perform the first inversion even when the second inversion is notpossible. After presenting the details of each of these inversions, we demonstrate their feasi-bility by applying them to invert the simulated spectra of a model dimer with a Markovianenvironment. We close with a consideration of the conditions under which state inversionwould be feasible for a natural light-harvesting system, the FMO complex of green sulfurbacteria.

4.2 Recipe for pump-probe spectroscopyWe begin by presenting the specific theoretical formalism for pump-probe spectroscopy thatwe propose to invert. The measured signal in any 3rd-order spectroscopy experiment, includ-ing pump-probe, is a function of the 3rd-order polarization [98]. This 3rd-order polarizationdepends on three interactions between the applied fields and the sample, with the time-ordering of these interactions enforced by time delays of the pulses and by looking at thesignal emitted in a particular phase-matched direction. For a pump-probe experiment, thefirst two interactions happen with the same pulse, the pump, and the last interaction is withthe probe pulse. The phase-matched condition is that the signal is observed in the directionof the probe. Based on this phase-matched geometry and the response function formalism[1] (see Appendix 4.7.1), we can combine the allowed time orderings to write the nonlinearpolarization for a pump-probe experiment under the rotating wave approximation as

P (3)(t) =

∫ ∞0

dt3RPP(t3, ρ(2)PP(t− t3))E+

pr(t− t3), (4.1)

in terms of the pump-probe response

RPP(t, ρ(2)PP) =

i

~Tr[µ(−)G(t)V (+)ρ

(2)PP

]. (4.2)

This pump-probe response is of identical form to the linear response function [98], but withthe electronic ground-state density matrix ρ0 replaced by the second-order contribution to the


density matrix ρ(2)PP that contributes to the phase-matched signal observed in a pump-probe

experiment (that is, with signal wave-vector kS = kpr). The quantities E+pu(t) and E+

pr(t)denote the complex envelopes of the pump and probe pulses, respectively, with E− ≡ (E+)∗.The dipole operators µ(−) =

∑n dnan and µ

(+) = (µ(−))†, with an as the annihilation operator

for an electronic excitation on pigment n and dn the corresponding dipole moments. TheLiouville space operator G(t) is the retarded material Green function for evolution for timet and V (±) · ≡ [µ(±), ·]. Formally, the portion of the second order contribution to the densitymatrix which contributes to the signal is given by

ρ(2)PP(t) =

(i

~

)2

2∑±

∞x

0

dt2dt1G(t2)V (±)G(t1)V (∓)ρ0E±pu(t− t2)E∓pu(t− t2 − t1). (4.3)

In deriving Eqs. (4.1–4.3), we employed the rotating wave approximation (accurate for res-onant excitation [1]) and neglected those terms from the double-quantum-coherence contri-bution (kS 6= kpr). Accordingly, we can safely neglect the possibility of multiple excitationsin the calculation of ρ(2)

PP. In Appendix 4.7.2 we prove that the excited state portion of ρ(2)PP

is both equal to the excited state portion of the full density matrix and is itself a valid (butunnormalized) density matrix.

The core of our proposed quantum state tomography is the sequential inversion ofEqs. (4.1–4.3). The remainder of this section discusses additional details relevant to simu-lating experimental signals to test our inversion procedure. We emphasize that these expres-sions hold under the very general conditions, requiring only the rotating wave approximation,that all applied fields are weak and negligible overlap between pump and probe pulses. Noassumptions were made concerning the shapes of these pump and probe pulses. Our decom-position here is similar to the window-doorway picture for the pump-probe signal [98], buthere we have separated out the influence of the control fields, even when not in the impulsive“snapshot” limit. Related expressions in terms of a convolution of pump and probe compo-nents have been shown to facilitate analysis of pump-probe experiments with shaped probes[112].

4.2.1 Detection scheme and probe convolution

In a typical pump-probe experiment, the probe pulse has a fixed time-envelope, subject toa variable delay time T between the two pulses. Accordingly, we may substitute E+

pr(t) =Epr(t − T ). Likewise, experimental signals are most directly interpreted in the frequencydomain, so we now consider the Fourier transform of the nonlinear polarization, P (3)(ω) =∫dt eiω(t−T )P (3)(t), calculated relative to the probe delay T . We can also write the pump-

probe response in the Fourier domain, RPP(ω, ρ(2)PP) =

∫dt eiωtRPP(t, ρ

(2)PP). In terms of these

quantities in the frequency domain with the explicit probe delay T , we can then replaceEq. (4.1) with a one-dimensional convolution,

P (3)(ω, T ) =

∫ ∞−∞

dτ RPP(ω, ρ(2)PP(τ))Epr(τ − T )eiω(τ−T ). (4.4)


To obtain this relation, we substituted t3 = t−τ and extended the lower limit of the integralin Eq. (4.1) to −∞, because by definition G(t) = 0 for t < 0. We cannot simply turn thisconvolution into a multiplication by taking the Fourier transform of these quantities withrespect to T , because for small or negative delay times T , there are contributions from signalswhere the pump does not necessarily interact before the probe. If RPP(ω, ρ

(2)PP(τ)) does not

vary appreciably over the duration of the probe pulse, then the equation above is the Fouriertransform of the probe field envelope, so we can approximate

P (3)(ω, T ) ≈ Epr(ω)RPP(ω, ρ(2)PP(T )). (4.5)

In the limit of a completely impulsive probe, Epr(t) ≈ E0δ(t) and thus Epr(ω) is constant,so the nonlinear polarization and the pump-probe response are equal up to a constant ofproportionality.

We measure the non-linear polarization P (3)(t) by detecting the corresponding signal fieldES(t) ∝ iP (3)(t) [98]. Here we consider heterodyne detection, either with the probe pulse asin a standard “self-heterodyned” pump-probe setup, or with a separate local-oscillator (LO)pulse. The use of a separate local-oscillator is possible in a transient-grating setup, in whichthe pump pulse is replaced by two otherwise identical pumps with different wavevectors k1

and k2, so that the signal wavevector kS = −k1+k2+k3 does not match the probe wavevectork3. Mathematically, this transient-grating signal yields the same non-linear polarization asin pump-probe, although it raises experimental complications by requiring phase-stabilitywith an additional pulse. In heterodyne detection, the absolute value squared of the sum ofthe signal and local-oscillator (or probe) fields can be spectrally dispersed and measured inthe frequency domain [131]. Typically, the signal field is much smaller than than of the localoscillator, so upon subtracting away the local oscillator contribution, the measured signalS(ω) is proportional to Re[ES(ω)E∗LO(ω)], and thus

S(ω, T ) ∝ Im[P (3)(ω, T )E∗LO(ω)]. (4.6)

This equation is a multiplication in the frequency domain. Hence it is a convolution, andtakes on similar form to Eq. (4.4) when expressed in the time-domain. In the pump-probesetup, ELO = Epr, so the signal for a fast probe given by inserting Eq. (4.5) yields

S(ω, T ) ∝ |Epr(ω)|2ImRPP(ω, ρ(2)PP(T )). (4.7)

In this case, the signal only depends on the imaginary (absorptive) part of the non-linearpolarization P (3) and the pump-probe response. In the alternative transient grating setup,as long as one is still in the limit of a fast probe, applying a π/2 phase shift to the nowdistinct local oscillator pulse allows for obtaining the real (dispersive) part of the pump-proberesponse function in a similarly direct manner [131]. More generally, heterodyne detectionwith and without a π/2 phase shift allows for obtaining both real and imaginary parts ofthe non-linear polarization, respectively.


4.2.2 Pump-probe response function

To isolate the effect of the probe, the pump-probe response function given by Eq. (4.2) canbe written as

RPP(t, ρ(2)PP) = Tr

[P(t)ρ

(2)PP

], (4.8)

with the pump-probe response operator P(t) defined as

P(t) =i

~µ(−)G(t)V (+). (4.9)

When inserted in Eq. (4.8), the action of P(t) is equivalent to the action of i/~[µ(−)(t), µ(+)(0)],where µ(±)(t) denotes µ(±) in the Heisenberg picture. This is similar but not equivalent toa family of quantum measurements [103] parametrized by the continuous time variable t (orfrequency ω in the Fourier domain), since RPP can be complex valued. Accordingly, thepump-probe response can be interpreted as the projection of ρ(2)

PP onto P(t), where these areviewed as vectors in Liouville space [98],

RPP(t, ρ(2)PP) =

⟨⟨P(t)

∣∣∣ρ(2)PP

⟩⟩. (4.10)

Individual components 〈〈P(ω)|α〉〉 of the pump-probe response operator are equivalent to thespecies associated spectra of the state |α〉〉 [143].

In most spectroscopy experiments, the signal is an ensemble measurement summed overall possible molecular orientations and static disorder of Hamiltonian parameters (inhomo-geneous broadening). Accordingly, the pump-probe response in Eq. (4.2) should be replacedby its average over molecular orientations and static disorder. The orientational averagecan be handled elegantly using the expression for the pump-probe response in Eq. (4.8): inthe magic angle θ ≈ 54.7◦ (MA) relative polarization configuration between the pump andprobe pulses [65], the quantities P(t) and ρ(2)

PP can simply be replaced by their independentisotropic averages, ⟨

RPP(t, ρ(2)PP)⟩MA

= Tr[⟨P(t)

⟩iso

⟨ρ

(2)PP

⟩iso

]. (4.11)

By virtue of the properties of isotropically averaged tensors [38], these independent isotropicaverages are equal to the average of the quantities obtained from the xx, yy and zz configu-rations. In contrast, the ensemble average over static disorder cannot be factorized this wayin general, because under static disorder the pump-probe operator and density matrix arecorrelated, and altering the system Hamiltonian (e.g., to shift transition energies) changesboth quantities systematically.


4.3 Inversion protocols

4.3.1 Deconvolution of the pump-probe signal

The first stage of our inversion protocol is a double-deconvolution to determine the complexvalued pump-probe response function RPP(T, ρ

(2)PP) from the results of a series of heterodyne

measurements, i.e., the signal S(ω, T ). We need such a double-deconvolution procedurebecause the results of heterodyne detection depend on a (trivial) convolution over the non-linear polarization, which in turn depends on a convolution over the response function [seeEqs. (4.4) and (4.6)]. Since the excited state density matrix is entirely contained in thepump-probe response function (Appendix 4.7.2), this inversion retains all information aboutthe quantum state. However, it is not immediately clear that the real (dispersive) part ofthe response function contains useful information independent of the imaginary (absorptive)part, which is the portion measured by usual pump-probe experiments.

Inverting the signal to obtain the pump-probe response function is a non-trivial butimportant task, since, as pointed out above, the signal is directly proportional to the responseonly when the probe pulse is much faster than all energy transfer dynamics. Such pulses canbe difficult to realize experimentally. The need for a full inversion to obtain the responsefunction is particularly relevant for understanding experiments which show fast oscillationsdue to quantum beats, whether these are of electronic, vibrational or mixed origin. In suchcases, the fast probe assumption of Eq. (4.5) is not valid. We shall refer to the use ofthis approximate description for inversion as the “naive” approach. In contrast, a propertreatment of this inverse problem would attempt to undo the convolution in Eq. (4.4).

To address this challenge, we suggest the use of standard deconvolution techniques [60]based on general-form Tikhonov regularization (also known as ridge regression), which wedescribe in detail in Appendices 4.7.3 and 4.7.4. The response function can then be ob-tained from two sequential 1D deconvolutions. First, we invert the measured signal S(ω, T )recorded at each choice of delay time T to obtain the nonlinear polarization P (3)(ω, T ). Therelationship between these signals is simple multiplication by the probe (or local oscillator)field in the frequency domain, so this step only uses the deconvolution to smooth the re-construction along the ω-axis. Second, we invert the nonlinear polarization P (ω, T ) witha one-dimensional deconvolution for each fixed value of ω to obtain the reponse functionRPP(ω, ρ

(2)PP(T )). For this inversion, we only use experimental data with the delay between

pump and probe pulses long enough so that we can ignore pulse overlap effects. Otherwise,we would be including non-pump-probe contributions to the signal. However, we neverthelessalso reconstruct the pump-probe response at shorter times in order to appropriately handleboundary conditions, since the probe convolution means that these values for the responsefunction contribute to the nonlinear polarization inside our region of interest.

This first stage in the inversion of pump-probe experiments requires only the detectionresults, i.e., the signal S(ω, T ), and an excellent characterization of the probe and localoscillator fields. No system information is required at all. Likewise, we have sacrificedno information from our measurement about the internal system information, including its


quantum state. Thus in principle this stage can be performed with high accuracy for anysystem, no matter how complex its internal degrees of freedom.

4.3.2 Obtaining the quantum state

The second step to complete the state tomography is to invert the pump-probe responsefunction RPP(ω, ρ

(2)PP(T )) to obtain the quantum state ρ(2)

PP(T ). This is certainly the harderstep, since it requires the ability to construct the pump-probe response of arbitrary states.The necessary information is contained in the pump-probe operator P(t) given by Eq. (4.9);calculating this requires both the transition dipole moments and a model for dynamics ofthe 1-exciton coherences between the probe and signal interactions. Essentially the sameinformation is necessary to implement proposed algorithms for quantum process tomography[157, 156]. However, we emphasize that we do not need to know the nature of the initial statecreated by the pump pulse nor any details of the energy transfer dynamics in the 1-excitonsubspace. The lack of required microscopic dynamical information is significant, since exactenergy transfer dynamics are non-trivial to calculate from first principles [106].

Here we consider a simple protocol for state tomography, based on an assumed modelfor calculating the pump-probe response. It is by no means the only such possible statetomography protocol: we choose it because it is straightforward to implement, and turns outto be relatively robust to imperfections such as static disorder. To perform the inversion, wepropose to extract an estimate of the excited-state electronic density matrix ρe(τ) from theestimated response function RPP(ωi, τ) at that delay time τ , for each frequency ωi matchingthe single-exciton transition energies. The relationship between the vector of pump-proberesponse measurements and the density matrix elements at any fixed time delay is linear[see Eq. (4.10)], so as long as this map is non-singular, we can solve for the density matrixby simply applying the matrix inverse to the vector formed by these estimated responsefunction points. It is possible that in some circumstances this reconstruction will not yielda valid density matrix, since we did not include the constraint that the reconstruction bepositive semi-definite. In this case, then a best estimate to minimize the mean-squared-errorof the reconstruction should be obtained using techniques based on maximum likelihood [77],although we do not encounter this issue for the examples we consider in this paper.

An additional important practical step is the choice of Liouville space in which the ex-tracted state lies. Our results so far hold for transition dipole operators and time evolutionwithout any particular restrictions concerning electronic vs vibrational states or the Hamil-tonian we use to describe our system. However, as a practical matter for excitonic energytransfer in light harvesting systems, we are most interested in determining the electronic de-gree of freedom. The electronic portion of ρ(2)

PP has useful structure: namely, it only includesnonzero elements in two blocks, the 0- and 1-excitation subspaces. We denote the projectionof the density matrix ρ onto these subspaces by ρg and ρe, respectively. There is only oneelectronic state in the 0-excitation subspace (the ground state g), so the electronic portionof ρg must be in that state, |g〉〈g|. In the Markov limit, or for delay times much longer thanthe bath relaxation time, the vibrational portion of ρg will be in thermal equilibrium ρBeq.


These facts determine ρg, up to a constant of proportionality: the ground state population.Because total probability is conserved in the process of laser excitation, Tr ρ

(2)PP = 0, so the

ground state population is related to the excited state population by Tr ρ(2)g = −Tr ρ

(2)e .

Accordingly, we can write

ρ(2)PP = −

(|g〉〈g| ⊗ ρBeq

)Tr[ρ(2)e

]+ ρ(2)

e . (4.12)

In this case, the pump-dependence in the pump-probe signal [Eq. (4.3)] is entirely containedin the excited state portion of the density matrix. Since for weak fields ρe ≈ ρ

(2)e , with

Eqs. (4.10) and (4.12) we have a linear map from any excited state density matrix ρe tothe corresponding pump-probe response. For a system with n electronic states, we canparameterize this unnormalized density matrix in terms of a linear combination of n2 realparameters [145], since the excited state density matrix is positive (see Appendix 4.7.2) andthus Hermitian.

A brief discussion of the scalability of this approach is in order. Based on the real andimaginary parts of the pump-probe response function in the magic angle configuration, ourstate tomography protocol in principle has 2n independent real parameters from which toextract the n2 real parameters (including normalization) necessary to describe an arbitraryexcited state density matrix of n electronic states [145]. Accordingly, we cannot necessarilyexpect this procedure to scale beyond a dimer (n = 2), for which we numerically demon-strate the success and stability of this inversion procedure in the next section. The recentlyproposed quantum process tomography algorithm based on peak and cross-peak amplitudesin 2D spectroscopy [156] has similar scaling difficulties. It requires determining n4 − n2 realparameters in the process matrix from at most 12n2 possible experimental measurements:the real and imaginary signals, n coherence and n rephasing frequencies, at most 3 indepen-dent polarization configurations and 2 phase-matched geometries. These estimates, however,hold only for this specific approach and with a randomly oriented ensemble. Oriented orsingle molecule measurements offer a much larger number of independent polarization mea-surements, a point we will return to Section 4.5.

4.4 Example: dimer modelTo understand in more detail how the quantum state determines the pump-probe signal, weconsider the case of the signal for a dimer of coupled pigments. In general, we can write aneffective Hamiltonian for the electronic excited states of a dimer in the form

Hel = E1a†1a1 + E2a

†2a2 + J(a†2a1 + a†1a2). (4.13)

The terms E1 and E2 are the transition energies of sites 1 and 2, and J is the pigment-pigmentcoupling energy. We restrict the system to at most one excitation on each site, so our statespace is spanned by the set {|g〉, |e1〉, |e2〉, |f〉}, denoting the ground state, excitation of thefirst or second site, and excitation of both sites. We further assume the usual linear coupling


to a bath of phonons. Details of the bath are specified below. The electronic part of thisHamiltonian can be diagonalized by applying a unitary rotation U to the single-excitationsubspace, given by

U =

[cos θ sin θ− sin θ cos θ

], (4.14)

where we defined the mixing angle θ = 12

arctan(2J/∆) with ∆ = E1 − E2. These singleexcitation eigenstates are denoted |α〉 and |β〉, The transition dipole moments for eachpigment are d1 and d2, oriented with relative angle φ.

4.4.1 Analytical calculation of pump-probe response

To begin, we choose a parametrization for the excited state density matrix of a dimer. Ingeneral, any valid density matrix for a two-level system can be written in any basis inform 1

2(I + r · σ), in terms of the Pauli matrices σ = {σx, σy, σz} and the Bloch vector

r = {r1, r2, r3}, with ri real and |r|2 ≤ 1 [103]. This can be straightforwardly generalized tounnormalized density matrices by adding the normalization r0 and defining σ0 = I, in whichcase the set of valid but unnormalized states are those that can be written in the form 1

2(r·σ),

where r is now the 4-dimensional vector {r0, r1, r2, r3} with constraints r21 + r2

2 + r23 ≤ r2

0 andr0 > 0. We will use these four real parameters to parametrize the excited state electronicdensity matrix ρe for our tomography protocol, since it has population r0 � 1. Usingthis representation, the total second-order correction to the electronic density matrix fromEq. (4.12) is given by

ρ(2)PP = −r0|g〉〈g|+

r · σ2

. (4.15)

For convenience, we suppose the state is written in terms of the eigenbasis expansion ofthe excited states {|α〉, |β〉}, so the parameters r1 and r2 correspond to excitonic coherencesand r3 corresponds to the balance of population between excitonic states. In practice, theexperimental signal is only known up to a constant factor, so we can only hope to be ableto reliably determine the normalized excited-state density matrix, given by the usual Bloch-vector elements {r1/r0, r2/r0, r3/r0}.

It is now straightforward (if tedious) to write down the exact pump-probe response func-tion in terms of microscopic parameters. For illustrative purposes, we do so here for a dimerwith a Markovian bath described by Redfield theory in the secular approximation [93]. Thetime-evolution contained directly in the pump-probe response function is for times followingthe probe interaction, so the relevant part of the system density matrix for this evolutionincludes coherences between ground and singly excited states and between singly and dou-bly excited states. In secular-Redfield theory, coherences in the excitonic basis only evolvewith exponential decay, G(T )|a〉〈b| = e−γabTΘ(T )|a〉〈b|, where G(T ) is the retarded materialGreen function denoting evolution for time T , Θ is the Heaviside step function, and γab is


some complex number with positive real part. Since the formulas for the response functionwill accordingly be most compact in the exciton basis, we consider the excitonic transitiondipole moments given by,

µgα = µ1 cos θ + µ2 sin θ (4.16a)µgβ = −µ1 sin θ + µ2 cos θ (4.16b)µαf = µ1 sin θ + µ2 cos θ (4.16c)µβf = µ1 cos θ − µ2 sin θ, (4.16d)

with µi = diai, where di is the component of the dipole-transition vector parallel to the probepolarization. For convenience, we define fα and f ′α to denote the Fourier transform of thetime evolution operator that leads to a peak in the pump-probe spectrum at frequency ωα,with decay constant γ or γ′, where the prime indicates the decay constant for the transitionbetween the 1- and 2-exciton manifolds instead of between the 0- and 1-exciton manifolds:

fα =1

i(ωα − ω)− γ, f ′α =

1

i(ωα − ω)− γ′. (4.17)

We define fβ and f ′β analogously, for the components peaked at ωβ.Using Eq. (4.10), the calculation of the pump-probe response function for an arbitrary

state is determined by the vectorized version of the pump-probe operator, 〈〈P(ω)|. Forthis dimer problem, we define the pump-probe bra vector such that RPP = 〈〈P|r〉〉, where|r〉〉 = r = {r0, r1, r2, r3}. Such a relation still holds upon substitution of |r〉〉 for ρ(2)

PP, sincethe relation between the two given by Eq. (4.15) is linear. With this convention, evaluatingthe pump-probe operator in Eq. (4.9) for this dimer model described by secular-Redfieldtheory yields the general result (see Appendix 4.7.7),

〈〈P| ∝

−3µ2

gαfα + µ2βff

′α − 3µ2

gβfβ + µ2αff

′β

−µgαµgβ (fα + fβ) + µαfµβf(f ′α + f ′β

)−i[µgαµgβ (fα − fβ) + µαfµβf

(f ′α − f ′β

)]−µ2

gαfα − µ2βff

′α + µ2

gβfβ + µ2αff

′β

T

, (4.18)

This equation holds for each single molecule that would contribute to the pump-probe signal.We can also calculate the exact isotropic average of Eq. (4.18) over an ensemble of randomlyoriented molecules. In terms of the original Hamiltonian parameters, it is given by

〈〈〈P|〉iso ∝

(cos2 θ + δ2 sin2 θ

)(f ′α − 3fα) +

(sin2 θ + δ2 cos2 θ

)(f ′β − 3fβ) + δ sin 2θ cosφ

(−f ′α + f ′β − 3fα + 3fβ

)− 1

2

(δ2 − 1

)sin 2θ

(f ′α + f ′β + fα + fβ

)+ δ cos 2θ cosφ

(f ′α + f ′β − fα − fβ

)i[12

(δ2 − 1

)sin 2θ

(f ′α − f ′β + fα − fβ

)+ δ cos 2θ cosφ

(−f ′α + f ′β + fα − fβ

)]−(cos2 θ + δ2 sin2 θ

)(f ′α + fα) +

(sin2 θ + δ2 cos2 θ

)(f ′β + fβ) + δ sin 2θ cosφ

(f ′α + f ′β − fα − fβ

)T ,

(4.19)

where θ is the excitonic mixing angle, δ = |d2|/|d1| is the ratio of the two site transitiondipole moments and φ is the angle between them. Note that neither of these equations


includes the effects of static disorder, which could be accounted for by averaging the pump-probe response function over each member of the ensemble. Formally, it does not suffice toseparately average 〈〈P |, since under static disorder the state |r〉〉 also varies in correlated way(see Sec. 4.2.2).

Equation (4.19) makes it possible to determine some cases in which solving for the isotrop-ically averaged state cannot possibly be successful, regardless of the exact inversion protocol.We can identify these cases because successful inversion requires that the elements of 〈〈〈P|〉isobe linearly independent. For example, in the homodimer case with both pigments fixed tohave the same transition energies (θ = π/4 or θ = 3π/4) and equal transition-dipole momentmagnitudes (a = 1), the pump-probe signal does not depend on the coherence terms, soit will be impossible to determine them (r1 and r2). Likewise, the coherence terms do notcontribute if the transition dipole moments have identical magnitude (δ = 1), and eitherthese are oriented perpendicularly (cosφ = 0) or there are matching dephasing rates for the0-1 and 1-2 coherences (fα = f ′α and fβ = f ′β, as occurs in the high-temperature limit). AsYuen-Zhou et al. found for the same dimer model [157], quantum process tomography alsofails under similar but not identical conditions.

4.4.2 Numerical example

For a numerical example, we consider the dimer model used in a prior investigation ofquantum process tomography [157]. We model excitation by a resonant 40 fs full-width-at-half-maximum (FWHM) pump centered at 12 800 cm−1. The parameters in the electronicHamiltonian are E1 = 12 881 cm−1, E2 = 12 719 cm−1 and J = 120 cm−1, and the experimentis performed on an ensemble with normally distributed static disorder of standard deviation40 cm−1 added to each site energy. The transition dipole moments are fixed with ratioδ = |d2/d1| = 2 and orientation angle φ = 0.3. Each pigment is assumed to be coupledto an independent bath of phonons, with spectral density of the form J(ω) = λ

ωcωe−ω/ωc

with ωc = 120 cm−1 and λ = 30 cm−1. The bath is assumed to be at thermal equilibrium atT = 273K and is modeled by secular Redfield theory [93], including only the real (dissipative)part of the Redfield tensor.

To simulate an experimental dataset, we first calculate the non-linear polarization P (3)(ω, T )for a probe of the same shape as the pump pulse, on a grid of 181 probe frequencies ω (in-tervals of 3.33 cm−1 between 12 500 cm−1 and 13 100 cm−1) and 140 central time-delays Tbetween pump and probe pulses (intervals of 6.81 fs between 50 fs and 1 ps). From this non-linear polarization, we then calculate the results of a hypothetical heterodyne detection witha local oscillator matching the probe pulse, with and without a π/2 phase shift. Finally, weaccounted for noise in detection by including additive noise with uniformly random phase andamplitude drawn from a standard deviation with width equal to 10−2 times the maximumamplitude over all delay times and frequencies of the heterodyne detected signal S(ω, T ).These simulated measurements, generated for comparison both with and without detectionnoise, are shown in Fig 4.1, together the response function from which they are calculated.


12500

12600

12700

12800

12900

13000

13100

ωα

ωβ

200 400 600 800 100012500

12600

12700

12800

12900

13000

13100

200 400 600 800 1000

ωα

ωβ

Delay time τ (fs)

Prob

e fr

eque

ncy ω

(cm−

1)

Het

erod

yned

sign

alR

espo

nse

func

tion

Absorptive part Dispersive part

Figure 4.1: Absorptive (left) and dispersive (right) parts of the pump-probe response functionRPP(ω, ρ

(2)PP(τ)) (top) and the corresponding heterodyne detected signal S(ω, τ) (bottom) for

our dimer model system. The dashed line indicates the two exciton transition energies in thissystem. Only the absorptive part (left) is revealed directly by a pump-probe experiment.Obtaining the dispersive part (right) requires a transient grating setup with heterodynedetection, as described in Sec. 4.2.1.

4.4.3 Response function inversion

Figure 4.2 illustrates the performance of the completed double Tikhonov regularizationbased deconvolution algorithm for typical noisy and noise free examples of our test prob-lem. We compare with the “naive” approach of assuming that the probe is impulsive andusing Eq. (4.5) to obtain the response function by simply dividing the signal by absolutevalue squared of the probe field |Epr(ω)|2. Table 4.1 summarizes the results of the simulatedinversion for the noise free case and 1000 such noisy examples. In addition to the doubleTikhonov and naive methods, we also consider the alternatives of substituting the naiveapproach individually for each of the two Tikhonov steps. Recall that in the first stage ofthe inversion (S → P (3)), the Tikhonov regularization serves only to smooth the data. It isnot surprising then that Table 4.1 shows that the specific method chosen for this first stage(i.e., naive or Tikhonov) makes no difference for the noise free case. In the second stage


0 200 400 600 800 1000Delay time τ (fs)

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

Pum

p-pr

obe

resp

onse

Rpp(ω

α,τ

)

Naive inversionTikhonov inversionOriginalProbe envelope

(a)

12500

12600

12700

12800

12900

13000

13100

200 400 600 800 100012500

12600

12700

12800

12900

13000

13100

200 400 600 800 1000

10−10

10−8

10−6

10−4

10−2

Delay time τ (fs)

Prob

e fr

eque

ncy ω

(cm−

1)

Tikh

onov

inve

rsio

nN

aive

inve

rsio

n

Noisy No noise

Error

(b)

Figure 4.2: (a) Example reconstruction of the pump-probe response at fixed probe-frequencyωα for an instance of the high-noise test problem. (b) Errors in the estimated pump-proberesponse obtained by the direct and Tikhonov inversion methods for a single example of thelow and high noise test problems. The error is given by the absolute value squared of thedifference between the estimated and actual response function, |Rpp(ω, τ)−RPP(ω, τ)|2.


Noise S → P (3) P (3) → RPP RMSE Improvement10−2 Naive Naive 12.12± 0.07 —10−2 Tikhonov Naive 7.78± 0.05 1.6± 0.010−2 Naive Tikhonov 3.34± 0.08 3.6± 0.110−2 Tikhonov Tikhonov 0.98± 0.06 12.5± 0.70 Naive Naive 7.110 —0 Tikhonov Naive 7.110 1.00 Naive Tikhonov 0.005 13010 Tikhonov Tikhonov 0.005 1301

Table 4.1: Summary of deconvolution performance over 1000 instances of simulated experi-mental noise. RMSE (root-mean-squared-error) is given by the sum of the absolute difference

between the estimated and actual response functions,(∑

ω,τ |Rpp(ω, τ)−RPP(ω, τ)|2)1/2

.Improvement is the multiple of the reduction in RMSE compared to the naive approach.Uncertainties indicate one standard deviation in the empirical distribution.

(P (3) → RPP), the Tikhonov regularization also performs a deconvolution over the probeenvelope.

The results in Fig. 4.2 and Table 4.1 show that the Tikhonov based inversion is a clearimprovement over the naive approach, reducing the root-mean-squared-error (RMSE) by afactor of 12 for our noisy example and 1300 for our noise-free example. The noisy and noisefree examples allow us to observe that the Tikhonov regularizations remove two types oferrors inherent in the naive inversion: (1) errors from noisy measurements and (2) errorsassociated with the convolution of the pump-probe response over the finite probe duration.In the noisy case, both errors are large; in the noise free case, there are only errors from thesecond source. Clearly, reducing the experimental noise associated with measurement alonedoes not suffice to accurately estimate the pump probe response, as the double Tikhonovinversion of the noisy signal outperforms naive inversion of the noise free signal by a factorof 7 in RMSE.

As Figure 4.2 shows, the errors in the estimate of the response function are not uniformlydistributed, revealing structure relevant to our specific example and also to more genericsystems. The largest errors are associated with smallest delay times. This makes sense,since the smallest delay times are those at which at the response function (shown in Fig. 4.1)varies most rapidly. For almost any system, the pump-probe spectrum will change fastest atshort delay times, but this is especially true for our example system, where the pump-probesignal includes contributions from quickly oscillating coherences. The Tikhonov estimatesface an additional stability challenge at short delays times, since, as discussed above, thereconstruction cannot use measurements from the pulse overlap regime.


4.4.4 State tomography

Since we have demonstrated that the first, response function inversion can be performed withvanishing error, we now consider inverting the exact pump-probe response function to obtainthe state of our model dimer. Despite the presence of static disorder in our example, we usethe factorization of the response function in Eq. (4.11). We are obliged to do so even thoughstrictly speaking the relationship does not hold, because the alternative of reconstructingthe density matrix for each member of the ensemble from a bulk measurement is unrealistic.Accordingly, even without adding noise associated with the measurement, when carried outfor an ensemble, our inversion faces potential stability issues because of the static disorder.

The degeneracies and near degeneracies in Eq. (4.19) mean that for most Hamiltonianchoices our inversion algorithm can only robustly extract at most three of the four parametersnecessary to fully characterize the dimer excited state, since the reconstruction matrix willbe poorly conditioned. The condition number of a linear transformation gives a bound onthe multiplicative increase in the relative error after performing the linear transformation[59]. For our specific numerical example, the condition number drops from 3700 to 3.1 whenwe include only three parameters. One source of these stability issues for a dimer is evidentfrom Eq. (4.19): since our numerical example has well separated transition energies, themain contribution to the peaks in the dispersive part of the signal is to the imaginary partof the coherence term, r2. This leaves our inversion to recover three parameters (r0, r1 andr3) from the two peak amplitudes in the absorptive signal. Since recovering three unknownsfrom two equations is not possible, we need to fix one of these values in order to make theinversion stable. The obvious choice is to fix the normalization r0, since the total excited statepopulation should remain constant after the end of the pump pulse until spontaneous decay,on timescales approaching 1 ns for natural pigment-protein complexes [9]. To determine thenormalization, we solve for it at a moderately long delay time (e.g., τ = 10 ps) at whichpoint we can safely assume (at least under secular Redfield dynamics) that the real part ofthe coherence r1 → 0, but very few excitations have been lost. If these timescales are noteasily separable, then this normalization term could be fit to an exponential decay.

The results of applying this state tomography procedure to our numerical example withvarying levels of static disorder are shown in Fig. 4.3. The fidelity, ranging from 0 to 1,provides a numerical summary of the quality of the state reconstruction [103]. For the levelof static disorder chosen by Yuen-Zhou et al. (40 cm−1), the reconstruction [panel (a)] has aworst-case fidelity of 99.5% over delay times T shorter than 1ps, and an average-case fidelityof 99.9%. However, we see that both the worst-case and average-case fidelities drop sharplyas the static disorder is increased above this level [panel (b)], since our assumption that thepump-probe response can be factorized between the pump-probe projection and the secondorder density matrix becomes increasingly unrealistic.


0 50 100 150 200Static disorder (cm−1 )

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Fide

lity

Average caseWorst case

0.40.20.00.20.4

r 1

0.40.20.00.20.4

r 2

200 400 600 800 1000Delay Time (fs)

0.40.20.00.20.4

r 3

(a)

(b)

Figure 4.3: Results of quantum state tomography for our dimer test problem. (a) Original(solid) and reconstructed (dotted) values for each element of the Bloch state vector for thereconstruction with static disorder of standard deviation 40 cm−1. Normalization is omittedsince the state vector elements are rescaled such that r0 = 1 fixed for all times followinginitial excitation. (b) Worst- and average-case fidelities for the reconstructions ρe(τ) fordelay times τ in the range 50 fs to 1 ps as a function of the width (standard deviation) ofthe distribution of static disorder. Results are obtained from an ensemble average over 106

samples for each point.


4.5 Scaling to larger systemsCan state tomography scale to systems larger than an excitonic dimer? In particular, canwe apply it to precisely reveal the excitonic state in a natural light-harvesting system?Any scaling difficulties will be encountered in the second step of our inversion protocol, todetermine the quantum state from the response function, since the relationship betweenthe response function and measured signal does not directly depend on system parameters.Successful state tomography certainly requires both knowledge of how each density matrixelement contributes to measurements and appropriate conditions such that each element, atleast in principle, makes an independent and non-zero contribution. By construction, theseconditions were satisfied for our hypothetical dimer example. We found that the primarylimitation on solving for the state was ensemble disorder, which can in principle be avoidedby single molecule techniques. Now, to explore the limits of state tomography, we relax theseassumptions in order to consider the feasibility of state tomography in an actual protein-pigment complex.

As a model light-harvesting system, we focus on a monomer of the Fenna-Matthews-Olson (FMO) complex of green sulfur bacteria, which consists of 7 pigment molecules [2,126]. The FMO complex is a widely used model system for understanding photosyntheticenergy transfer and is thus is one of the best characterized natural protein-pigment complex.The crystal structure for the FMO complex is known, which combined with input from spec-troscopy experiments, has allowed for general agreement on an electronic Hamiltonian [2, 62].Because the arrangement of pigments is fairly disordered, each excited state in a monomerof the FMO complex is bright, although they overlap in the presence of homogeneous andinhomogeneous broadening. This is important, since full state tomography would certainlynot be possible on a system with multiple dark states, because no optical probes could revealthe distribution of energy among those states. However, typical of the situation for othernatural pigment-protein systems, there is little consensus on the magnitude of the staticdisorder or the spectral density of the electronic-vibrational coupling [142]. These difficultiesare compounded by the theoretical and computational challenge of modeling dynamics in asystem as large as FMO exactly for arbitrary system-bath interaction strength [75]. For ourconcrete example, we use the electronic Hamiltonian for FMO of Chlorobaculum tepidumfrom Ref. [2], with the spectral density and computational model of secular Redfield theorymatching those used in for the dimer example. This model includes only one electronicstate per pigment and assumes Gaussian distributed static disorder with standard deviation42.5 cm−1 (100 cm−1 FWHM).

Our formalism for the pump-probe response function allows us to place bounds on thefeasibility of any state tomography procedure, since the relationship between system in-formation and the resulting pump-probe spectra is entirely contained in the pump-proberesponse operator. By looking at the ensemble average of this operator, we implicitly con-sider inversion under the scenario that the average over static disorder can be factorizedbetween the pump-probe operator and the density matrix,. This assumption was successfulin the dimer example above when the magnitude of static disorder was not too large. As


discussed in Section 4.2.2, the pump-probe operator at each frequency can be interpreted asa Liouville space bra-vector 〈〈P(ω)|. Accordingly, it is possible to interpret the calculationof a pump-probe response as the act of applying the linear operator

P =

∫dω|ω〉〈〈P(ω)| (4.20)

to the state |ρe〉〉. We now consider the properties of the linear operator P in the limitof effectively continuous sampling of probe frequencies ω. To represent states in Liouvillespace, we use a basis set that allows us to represent each state with n2 real values, in termsof populations |n〉〈n| and coherences |n〉〈m|+ |m〉〈n| and i|n〉〈m| − i|m〉〈n|. This allows usto construct a real-valued version of the map P that takes real valued state vectors to realvalued spectra by concatenating the real and imaginary parts of P.

To begin, in Figure 4.4 we plot the elements of the absorptive (real) part of the pump-probe operator for our model of the FMO complex in the isotropic average (magic an-gle configuration). We represent the operator in terms of the species associated spectra〈〈P(ω)|α〉〉 of each exciton population and the real (α = |n〉〈m| + |m〉〈n|) and imaginary(α = i|n〉〈m| − i|m〉〈n|) part of each coherence, so that the pump-probe spectra of anyexcited state ρe is equal to the linear combination of the plotted spectra weighted by theindicated density matrix elements,

〈〈P(ω)|ρe〉〉 =∑α

〈〈P(ω)|α〉〉〈〈α|ρe〉〉. (4.21)

In addition to the unperturbed spectra, we also plot the range of possible spectra givencurrent uncertainty about the best fit parameters. We conservatively estimate the uncer-tainty in the electronic Hamiltonian by sampling over additive independent Gaussian noise ofwidth 20 cm−1 for each site energy and 10% of the value of each off-diagonal coupling. Thisuncertainty is in addition to the static disorder, which we leave with fixed magnitude. Themost striking feature of these spectra is that, at least in the isotropic average, the dominantcontribution to the pump-probe spectra is from the population terms. The smaller contribu-tion of most coherence terms, compounded by the already smaller values of the coherencesin the density matrix due to dephasing, explains why it is difficult to observe oscillationsdue to electronic coherence in pump-probe spectra [126]. Even for extremely precise mea-surements, the uncertainty in some of these species associated spectra suggests that ourcurrent Hamiltonian characterization does not suffice to reliably obtain most density matrixelements. Indeed, the dominance of the diagonal terms suggests that a practical scheme forpartial state tomography could consist of entirely ignoring the off-diagonal terms.

Another approach to estimating the feasibility of inversion for arbitrary states is to look atthe spectral properties of the operator P as revealed by the singular value decomposition, P =USV †, where U and V are unitary and S is diagonal with positive elements. In particular,we focus on the singular values σi, given by the diagonal elements of S in descending orderand normalized to the highest singular value σ1. To compare the feasibility of inversion


<ρ11 <ρ12 <ρ13 <ρ14 <ρ15 <ρ16 <ρ17

=ρ21 <ρ22 <ρ23 <ρ24 <ρ25 <ρ26 <ρ27

=ρ31 =ρ32 <ρ33 <ρ34 <ρ35 <ρ36 <ρ37

=ρ41 =ρ42 =ρ43 <ρ44 <ρ45 <ρ46 <ρ47

=ρ51 =ρ52 =ρ53 =ρ54 <ρ55 <ρ56 <ρ57

=ρ61 =ρ62 =ρ63 =ρ64 =ρ65 <ρ66 <ρ67

1.20 1.22 1.24 1.26 1.2880604020

02040

=ρ71 =ρ72 =ρ73 =ρ74 =ρ75 =ρ76 <ρ77

Am

plitu

de (a

rbitr

ary

units

)

Probe frequency (×104 cm−1 )

Figure 4.4: Species associated spectra, defined by the contribution of the marked densitymatrix elements to the pump-probe response, for the FMO complex at 77K (blue) and300K (red), obtained as the average of 1000 samplings over static disorder. Labels indicatethe contributing density matrix element in the excitonic basis. Shaded regions indicatecentral 95% confidence intervals obtained from 1000 additional samplings over Hamiltonianuncertainty, as described in the text.


0 10 20 30 40 50Index i

10-6

10-5

10-4

10-3

10-2

10-1

100

Sing

ular

val

ue σi/σ

1

Oriented (single molecule)Oriented (ensemble)Isotropic (abs. + disp.)Isotropic (abs. only)Isotropic (at 300 K)

Figure 4.5: Normalized singular values from a singular value decomposition of the real val-ued pump-probe map P given by Eq. (4.20) for the FMO complex under various conditions.The top line is from the spectra of a single monomer at 77K, from the combination ofmeasurements in all independent polarization configuration. Subsequent lines add addi-tional constraints, which apply cumulatively: ensemble measurement (over static disorder),the isotropic average of the signal, only the absorptive (real) part of the signal and finallyperforming the measurement at room temperature.

under various conditions, we plot these singular values in Figure 4.5. The singular valuesreveal significant information about the feasibility of an inversion: in general, inversion ismore feasible when the singular values σi decay more slowly [60]. For example, the conditionnumber, which gives an upper bound on the ratio by which the relative error can increase inan inversion, is equal to the ratio of the largest to the smallest singular values. In Figure 4.5,the condition number is one over the value shown for i = 49.

Figure 4.5 provides an indication of the relative significance of different experimentalconstraints insofar as they affect state tomography. We see two major changes that reducethe condition number of the inversion by around two orders of magnitude each. First, re-ducing temperature from 300K to 77K improves the conditioning because spectral featuresin the species associated spectra are sharpened. Second, changing from a randomly ori-


1 2 3 4 5 6 7

1234567

Isotropic x

x y z

y

z

(a)

(b)

Figure 4.6: Maximum amplitude over probe frequencies of the species associated spectra foran FMO monomer at 77K for (a) the isotropic average and (b) each independent polarizationconfiguration of the probe and local oscillator, including the ensemble average over staticdisorder. The labeling of each species matches that used in Figure 4.4: all entries includingand above the diagonal correspond to the real part of the matching density matrix element(in the excitonic basis), and all entries below the diagonal correspond to the imaginary part.The cartesian coordinates were chosen arbitrarily, matching those used in an assignment ofthe crystal structure.

ented (isotropic) to an oriented sample helps because in principle 9 times more independentmeasurements are possible than in the magic angle configuration, one for each combina-tion of x, y and z polarizations for the probe and signal interactions. This is illustratedin Figure 4.6, which plots the peak amplitudes of each of the species associated spectra.In some cross-polarization configurations, coherences give contributions of similar magni-tude to populations. Single molecule measurements do offer an important advantage overoriented ensembles that is not seen in this figure, since their analysis does not require anyassumption that the average over static disorder can be factorized between pump and probe.Finally, this figure suggests that the dispersive component of the signal, which as discussedin Section 4.2.1 requires a transient grating experiment, offers little additional informationcompared to the absorptive component that is provided directly by the pump-probe signal.

4.6 ConclusionsWhat information does an ultrafast spectroscopy experiment tell us about an excitonic sys-tem? How can we best design these experiments? We believe that a powerful way to answerthese questions is to treat ultrafast spectroscopy as a hierarchy of statistical inverse problems,


as we have demonstrated here for estimating the electronic excited state. By dividing ouranalysis into many smaller steps—from experimental signal to response function to excitedstate and eventually on to dynamics, equations of motion and Hamiltonian parameters—wecan see exactly how and where our models and experiments fall short. In this regard, ourapproach contrasts strongly with the established procedure of “forward simulation” for de-termining Hamiltonian parameters in complex excitonic systems by simultaneously fittingmany experiments with an assumed theoretical model for calculating spectra from first prin-ciples [2, 105]. Moreover, to identify the time-evolving excited state or another intermediatequantity in our approach, we do not need to introduce additional errors by recalculatingfrom first principles.

Our multi-stage inversion has clear extensions to more general non-linear spectroscopiesbeyond QST and pump-probe. As we describe in Appendix 4.7.5, we could apply essen-tially the same inversion procedure to a photon-echo experiment to determine the phase-matched component of the 2nd-order density matrix that contributes to the observed signal[Eq. (4.24)]. If it is possible to construct a full set of independent phase-matched initialconditions, our state tomography procedure could be used for process tomography, along thelines of a previous proposal [157]. More generally, this suggests a new indirect approach forprocess tomography: first, field information should be used to obtain the 3rd-order responsefunction (as outlined in Appendix 4.7.6); second, system information should be used to ex-tract the process matrix. Determining the response function as an intermediate quantityallows us to be sure we have obtained the maximum information from experiments beforeconsidering the harder theoretical problem of extracting system parameters, such as thestate, process matrix or underlying Hamiltonian. Such extensions will be pursued in futurework.

More immediately, our approach is particularly well suited to evaluating the benefitsof employing colored pulses or an ultrafast pulse shaper, because our formalism makes noassumptions about the shape of pump and probe pulses: they are only required not tooverlap in time. In contrast, prior proposals for process tomography relied on the assumptionthat interactions with laser pulses are much faster than the timescale of dissipative systemdynamics [157, 156]. Accordingly, we envision potentially using our scheme for verification ofstate preparation following shaped pump pulses [15, 24], scenarios for which single moleculesor oriented samples are similarly helpful. Indeed, pump-probe spectroscopy has been used toexperimentally verify ultrafast coherent control [115]. The deconvolution step in our inversionprotocol also made no assumptions about the shape of the probe pulse. Although ourextensive numerical simulations found no cases in which a shaped probe pulse was preferableto the corresponding time-frequency limited pulse, our inversion protocol can just as easilyobtain the non-linear response from, say, pump-probe measurements where the probe hasresidual chirp. This suggests the possibility of improving pulse characterization for bettertime resolution instead of or in addition to efforts to further compress pulses in time [112].

Finally, we point out that we have demonstrated state tomography in this work mostlyfor ensemble systems, including averaging over molecular orientations and static disorder.These features inevitably reduce the performance of the inversion. However, both oriented


and single molecule experiments may be possible in the near future, given recent advancesof preforming ultrafast spectroscopy on crystallized proteins [70] and the possiblity of usingnon-linear florescence measurements with phase-cycling [89] to scale non-linear spectroscopyto single molecules. For such single molecule experiments, we expect the present inversionwill provide a powerful analytical tool.

4.7 Appendices

4.7.1 Derivation of pump-probe signal

Rearranging the terms that result from inserting Eq. (1.13) into Eq. (1.12) yields Eqs. (4.1–4.3).

4.7.2 Positivity of ρ(2)e

We consider the system density matrix ρ in the presence of weak fields of strength ε � 1.Let ρ(n) denote the contribution to the density matrix of strength O(εn). Thus we can writeρ = ρ(0) + ρ(1) + ρ(2) + ρ(3) + O(ε4). Since the temperature is typically several orders ofmagnitude smaller than the electronic energy gap, the system starts in a tensor product ofthe electronic ground state |g〉〈g| and the equilibrium vibrational state ρBeq, ρ(0) = |g〉〈g|⊗ρBeq.

We are interested in the excited state portion of ρ(2)PP, the component of ρ(2) that con-

tributes to the phase-matched signal kS = kpr given by Eq. (4.3). We write this excitedstate density portion as ρ(2)

e = Qρ(2)PP, where Q denotes the projection onto the 1-excitation

manifold. Since the non-phase-matched components involve 2-excitation states, we also haveρ

(2)e = Qρ(2). This projection is given by Qρ = I1ρI

†1, where I1 =

∑m |m〉〈m| is the identity

operator restricted to the 1-excitation manifold and |m〉 denotes the state where only pig-ment m is excited. Since the map Q is written in the appropriate form and

∑n InI

†n = I, Q

is completely positive, with 0 ≤ TrQρ ≤ 1 [103].Moving from the ground state to the 1-excitation manifold requires at least two applica-

tions of the creation/annihilation operators contained in the dipole transition operators µ(±),and a dipole operator must be applied an even number of times. Accordingly, Qρ(n) = 0 forn = 0, 1, 3, which leaves Qρ = ρ

(2)e + O(ε4). Since we proved Qρ is positive and ρ(2) itself

is O(ε2), we have shown that ρ(2)e is positive, up to relative errors of O(ε2). Any positive

operator is a valid density matrix if it has trace one [103].

4.7.3 Tikhonov regularization

The convolution of the pump-probe response with the probe pulse in Eq. (4.4) and thenon-linear polarization with the local-oscillator in Eq. (4.6) are both particular cases of aFredholm integral equation of the first kind. An extensive literature exists on numerical


inversion of such equations, known as discrete inversion problems [60]. In general, the dis-cretization of such an integral equation can be written as

b = Ax+ ε, (4.22)

where b is the measured signal [e.g., the nonlinear polarization P (3)(ω, T )], x is the desiredquantity to obtain [e.g., the response function RPP(ω, T )], A is a linear operator representingthe integral equation with appropriate coefficients (determined here by the probe field Epr)and ε denotes some additive experimental error inherent in the data collection. The obvioussolution to estimating x from A and b is to calculate x = A−1b. However, in practice Amay not be invertible. This is the case for our inversion, since we ignore the experimentalsignal for times in the pulse overlap regime but still attempt to reconstruct the pump-proberesponse at those times, which guarantees that b has a lower dimensionality than x. Inaddition, the presence of even a vanishingly small amount of experimental noise ε makesexact least-squares minimization unsuitable; it will over-fit the noise component ε.

Accordingly, to calculate a robust estimate x of x we use general form Tikhonov regu-larization [60],

x = argminx

{||Ax− b||2 + λ2||Lx||2

}. (4.23)

Tikhonov regularization can be derived formally from the perspective of Bayesian inference,given normally distributed errors and priors [150]. It can be equivalently be expressed asthe linear least-squares problem, min ||[ AλL ]x − [ b0 ]||2, and thus the exact solution is givenby x = [ AλL ]+[ b0 ], where + denotes the Penrose-Moore pseudoinverse [60]. Ideally, the linearoperator L and (real) regularization parameter λ are chosen so that λ2||Lx||2 is an optimallyweighted penalty on undesirable features of the solution x, reflecting our prior knowledgeof the general form of x. Common choices of L include the identity matrix I and finite-difference approximations to the first or second derivative given by (D1x)n = xn− xn−1 and(D2x)n = xn+1 − 2xn + xn−1. We compare different techniques for selecting λ and L inAppendix 4.7.4.

There are a number of powerful techniques for calculating efficient approximate solutionsto Eq. (4.23), especially in cases where the linear operator A is structured [59], such as in ourcase, where A is a Toeplitz matrix. For several hundred time-delays or probe frequencies,we find that we can solve each deconvolution with Eq. (4.23) exactly and quickly (∼1 s on amodern CPU) by calculating the singular value decomposition of the matrix [ AλL ]. In princi-ple, it would be possible to solve both steps in the inversion of the pump-probe response in asingle two-dimensional Tikhonov regularization. Such 2D inversions are routinely performedin image processing [60], but would require slower, more approximate techniques than theexact solution we used here. Since we find significant improvement without invoking thesemore complicated methods, we do not use them here.


Noise Penalty Selection λopt Improvement10−2 I Exact 0.151± 0.007 2.3± 0.210−2 D1 Exact 0.340± 0.034 4.9± 0.710−2 D2 Exact 0.67± 0.11 6.2± 1.210−2 D2 GCV 0.78± 0.17 5.7± 1.210−2 D2 NCP 2.57± 0.36 2.3± 0.410−3 I Exact 0.049± 0.017 5.9± 0.510−3 D1 Exact 0.073± 0.010 30± 510−3 D2 Exact 0.112± 0.017 56± 1210−3 D2 GCV 0.103± 0.019 52± 1210−3 D2 NCP 0.449± 0.039 19± 2

Table 4.2: Regularization performance for different penalty operators and parameter selec-tion techniques for 1000 instances of random noise with relative magnitude 10−2 or 10−3.Numbers are the mean plus or minus one standard deviation. Improvement is the multiple ofthe reduction in mean-squared-error for the reconstructed response function using Tikhonovregularization over the error associated with the naive impulse-probe estimate.

4.7.4 Parameter selection for Tikhonov regularization

To perform deconvolutions using Eq. (4.23), we need to choose a procedure to select theregularization parameters λ and L. In practice, there are a wide variety of techniques formaking these selections and the best choice depends on the particular problem at hand [60].Here we compare the performance of different techniques on simulations matching the dimerproblem we analyze in Section 4.4.

To begin, we compared the performance of general form Tikhonov regularization with Lequal to I, D1 and D2, with λ chosen optimally so as to minimize the exact mean-squared-error ||x− x||2. A summary of reconstructions of RPP(ω, ρ

(2)PP(T )) for 1000 instances of low

and high noise is shown in Table 4.2. As a benchmark, we consider the ratio of the mean-squared-error from the Tikhonov estimate to the mean-square-error of our naive estimateRPP(ω, ρ

(2)PP(T )) ∝ P (ω, T )/Epr(ω), which holds in the limit of an instantaneous probe pulse

[Eq. (4.5)]. For our state inversion algorithm, reconstruction of the response function is mostimportant at frequencies matching the exciton transition energies, so we picked ω = ωα, thetransition frequency of the higher energy exciton state. We found qualitatively similar resultsfor ω = ωβ and other choices of ω as well. As Table 4.2 shows, with exact selection of thebest regularization parameter λ, we found consistently best performance with D2, the linearoperator approximating the second derivative of the response function with respect to thedelay time T . This is an intuitively reasonable choice, since plausible response functionsshould be smooth.

With the choice for L determined, we also need a realistic procedure for selecting theregularization parameter λ. In a true inversion problem the response function x is unknown,


so we cannot choose λ to minimize the exact mean-squared-error. There are a variety of stan-dard techniques for making this choice, with performance that can vary widely dependingon the problem being solved, so selection of an appropriate method requires more tests onsimulated data. We considered two such methods for L = D2: generalized cross-validation(GCV) and the normalized cumulative periodogram (NCP). We calculate the GCV error us-ing the exact pseudoinverse solution [150] and the NCP error by adding together the errorsfor the real and imaginary parts of the spectra [60]. We then minimize these error estimatesas a function of λ using a 1-dimensional search with the downhill simplex method [102].We also impose the additional restriction λ ≥ 5 × 10−11 to avoid convergence failures withour SVD implemention that we encountered when performing deconvolutions on noise-freesimulated spectra. The results, also shown in Table 4.2, show that GCV is the best choicefor our test problem, with performance nearly matching that of exact selection technique. Incontrast, NCP systematically overestimated the noise, as indicated by regularization param-eters about four times larger than the exact selection method. However, NCP still offered animprovement in the reduction of the mean-squared-error compared to the naive approach.

4.7.5 Alternative formulation

There are several obvious extensions or alternatives to the recipe described in section 4.2.For example, the exact same relations in Eqs. (4.1–4.3) hold for a general photon-echo (ornon-rephasing) experiment with two distinct pump pulses, except that in this situation thesum in Eq. (4.3) to determine the portion of ρ(2) that contributes to the signal should beremoved to leave only one of the two phase-matched contributions. Instead, the photon-echo(PE) signal depends on the second-order density matrix given by

ρ(2)PE(t) =

(i

~

)2 ∞x

0

dt2dt1G(t2)V (+)G(t1)V (−)ρ0E+2 (t− t2)E−1 (t− t2 − t1). (4.24)

However, unlike the case for the state ρ(2)PP that contributes to the pump-probe signal, the

excited state portion of ρ(2)PE is not necessarily either hermitian or equivalent to the total ex-

cited state density matrix. This makes its interpretation less clear but presents no additionaltechnical difficulties for our inversion procedure.

4.7.6 Triple deconvolution to obtain the full 3rd-order responsefunction

The deconvolution procedure demonstrated in Chapter 4 has an interesting application formore general 3rd-order spectroscopy experiments, beyond pump-probe. In particular, weenvision applying it to estimate the full 3rd-order response function R

(3)ks

(t3, t2, t1) from aseries of ultrafast measurements. This would formalize the standard interpretation of 2Dphoton-echo as snapshots of the 3rd-order response function, even in cases where the laser


pulses cannot be accurately described as fully impulsive, and allow for quantitative compar-isons between different experimental setups and theory independent of the particular lasersetup. Indeed, since this response function contains all possible system-information, suchdetermination of the third order response function would arguably constitute the “ultimate”3rd-order spectroscopy experiment, because all other possible third-order experiments couldbe calculated from numerical integration of the response function over the applied fields.

To illustrate this proposal, we rewrite the equation for the non-linear polarization givenby Eq. (1.12) in terms of the central times τi for each laser pulse i and the delay timesbetween each pulse, T3 = t− τ3, T2 = τ3 − τ2 and T1 = τ2 − τ1:

P(3)ks

(T3, T2, T1) =

∞y

0

dt3 dt2 dt1 R(3)ks

(t3, t2, t1)

× E(T3 − t3)E(T3 + T2 − t3 − t2)E(T3 + T2 + T1 − t3 − t2 − t1). (4.25)

Here we use the fixed laser envelope E(t) ≡ Ei(t + τi) for each pulse and assumed thatEi(t) is strictly real-valued in the rotating frame. We also assumed a definite time-orderingbetween each system-field interaction; in general, this equation should be replaced by a sumover all six possible time-orderings. In the impulsive limit E(t) ≈ δ(t), each integral dropsaway and we have R(3)

ks(T3, T2, T1) ≈ P

(3)ks

(T3, T2, T1). More generally, this equation providesa linear map from the response function to the non-linear polarization, which holds even ifall possible time-orderings are considered.

Because a typical 3D spectrum requires ∼ 102 measurements along each time axis, R(3)ks

is a ∼ 106 dimensional vector, and thus the transformation matrix for this linear relationwould have ∼ 1012 elements and be too large to fit in memory on most modern computers.To efficiently invert this relation, we write it as a series of three nested one-dimensionalconvolutions,

P(3)ks

(T3, T2, T1) =

∫dt3S2(t3, T3 + T2, T1)E(T3 − t3) (4.26)

S2(t3, T2, T1) =

∫dt2S1(t3, t2, T2 + T1)E(T2 − t3 − t2) (4.27)

S1(t3, t2, T1) =

∫dt1R

(3)ks

(t3, t2, t1)E(T1 − t3 − t2 − t1), (4.28)

in terms of the intermediate quantities S1 and S2. Since each of these relations specifies a setof one-dimensional convolutions, we should be able to independently invert each of them usingthe same generalized Tikhonov regularization technique that we demonstrated for Eq. (4.4).Note that this decomposition into a hierarchy of three convolutions requires a definite timeordering between the pulses. In fact, in the usual setup for 2D spectroscopy, it turns outthat only a definite time ordering between the second and third pulse is important; overlapbetween the first two pulses can be accounted for exactly by the convention that negativetimes t1 refer to the non-rephasing instead of the photon-echo direction. Fortunately, this


constraint of no pulse overlap between the second and third pulses is not insurmountable,since it restricts only the last convolution and thus can be dealt exactly as demonstrated inSec. 4.3.1, by restricting the deconvoluted signal to population times no smaller than someminimum delay.


4.7.7 Mathematica code for derivation in Sec. 4.4.1Derivation of analytical spectra of dimer [Section IV (A)]ü Initial conditions and operators in Liouville space:

m =

0 mga mgb 0

0 0 0 maf

0 0 0 mbf

0 0 0 0

;

r0 =

-r0 0 0 0

012Hr0 + r3L

12Hr1 - Â r2L 0

012Hr1 + Â r2L

12Hr0 - r3L 0

0 0 0 0

;

Gt =

0 0 0 0

fa 0 0 0

fb 0 0 0

0 fpb fpa 0

;

sitedipoles =8mga Ø Cos@qD d1 + Sin@qD d2, mgb Ø -Sin@qD d1 + Cos@qD d2, maf Ø Sin@qD d1 + Cos@qD d2, mbf Ø Cos@qD d1 - Sin@qD d2<;

ü Derivation of Eq. (18):S = [email protected] Hm!.r0 - r0.m!LLD êê FullSimplifyL

1

2II-3 fa mga2 - 3 fb mgb2 + fpb maf2 + fpa mbf2M r0 + H-Hfa + fbL mga mgb + Hfpa + fpbL maf mbfL r1 +

Â HHfa - fbL mga mgb + H-fpa + fpbL maf mbfL r2 + I-fa mga2 + fb mgb2 + fpb maf2 - fpa mbf2M r3M

Table@CoefficientList@2 S, 8ri<DP2T êê FullSimplify, 8i, 0, 3<D ê. 8Â Ø -Â< êê MatrixForm

-3 fa mga2 - 3 fb mgb2 + fpb maf2 + fpa mbf2

-Hfa + fbL mga mgb + Hfpa + fpbL maf mbf

-Â HHfa - fbL mga mgb + H-fpa + fpbL maf mbfL

-fa mga2 + fb mgb2 + fpb maf2 - fpa mbf2

Note the switched sign assigned to i because of the complex conjugate when turning this from a ket- to a bra-vector.

ü Derivation of Eq. (19):S2 = Collect@2 S ê. sitedipoles, 8d1, d2<D ê. 9d12 Ø 1, d22 Ø d2, d1 d2 Ø d Cos@fD=

fpa Cos@qD2 r0 - 3 fa Cos@qD2 r0 + fpb Sin@qD2 r0 - 3 fb Sin@qD2 r0 + Hfpa + fpbL Cos@qD Sin@qD r1 + Hfa + fbL Cos@qD Sin@qD r1 +

Â H-fpa + fpbL Cos@qD Sin@qD r2 - Â Hfa - fbL Cos@qD Sin@qD r2 - fpa Cos@qD2 r3 - fa Cos@qD2 r3 + fpb Sin@qD2 r3 +

fb Sin@qD2 r3 + d Cos@fD I-2 fpa Cos@qD Sin@qD r0 + 2 fpb Cos@qD Sin@qD r0 - 6 fa Cos@qD Sin@qD r0 + 6 fb Cos@qD Sin@qD r0 +

Hfpa + fpbL Cos@qD2 r1 + H-fa - fbL Cos@qD2 r1 + H-fpa - fpbL Sin@qD2 r1 + Hfa + fbL Sin@qD2 r1 +

Â H-fpa + fpbL Cos@qD2 r2 + Â Hfa - fbL Cos@qD2 r2 - Â H-fpa + fpbL Sin@qD2 r2 - Â Hfa - fbL Sin@qD2 r2 +

2 fpa Cos@qD Sin@qD r3 + 2 fpb Cos@qD Sin@qD r3 - 2 fa Cos@qD Sin@qD r3 - 2 fb Cos@qD Sin@qD r3M +

d2 Ifpb Cos@qD2 r0 - 3 fb Cos@qD2 r0 + fpa Sin@qD2 r0 - 3 fa Sin@qD2 r0 + H-fpa - fpbL Cos@qD Sin@qD r1 +

H-fa - fbL Cos@qD Sin@qD r1 - Â H-fpa + fpbL Cos@qD Sin@qD r2 +

Â Hfa - fbL Cos@qD Sin@qD r2 + fpb Cos@qD2 r3 + fb Cos@qD2 r3 - fpa Sin@qD2 r3 - fa Sin@qD2 r3M

TableBHCoefficientList@S2, 8ri<DP2T êê FullSimplifyL ê. :Cos@qD Sin@qD Ø1

2Sin@2 qD>, 8i, 0, 3<F ê. 8Â Ø -Â< êê MatrixForm

Ifpa - 3 fa + Hfpb - 3 fbL d2M Cos@qD2 + Ifpb - 3 fb + Hfpa - 3 faL d2M Sin@qD2 + H-fpa + fpb - 3 fa + 3 fbL d Cos@fD Sin@2 qD

Hfpa + fpb - fa - fbL d Cos@2 qD Cos@fD -12Hfpa + fpb + fa + fbL I-1 + d2M Sin@2 qD

-Â Hfpa - fpb - fa + fbL d Cos@2 qD Cos@fD +12Â Hfpa - fpb + fa - fbL I-1 + d2M Sin@2 qD

-Ifpa + fa - Hfpb + fbL d2M Cos@qD2 + Ifpb + fb - Hfpa + faL d2M Sin@qD2 + Hfpa + fpb - fa - fbL d Cos@fD Sin@2 qD

85

Chapter 5

Realistic and verifiable coherent controlof excitonic states in a light harvestingcomplex

SummaryWe explore the feasibility of coherent control of excitonic dynamics in light harvesting com-plexes despite the open nature of these quantum systems. We establish feasible targets forphase and phase/amplitude control of the electronically excited state populations in theFenna-Mathews-Olson (FMO) complex and analyze the robustness of this control. We fur-ther present two possible routes to verification of the control target, with simulations forthe FMO complex showing that steering of the excited state is experimentally verifiable ei-ther by extending excitonic coherence or by producing novel states in a pump-probe setup.Our results provide a first step toward coherent control of these systems in an ultrafastspectroscopy setup.

5.1 IntroductionThe control of atomic and molecular processes using coherent sources of radiation is a wellestablished experimental technique. Particularly successful have been implementations thataim to control the non-equilibrium dynamics of highly coherent quantum systems, e.g. inter-nal and external states of cold atoms [36], and ones that aim to control the equilibrium statesresulting from controlled mixed (coherent and incoherent) dynamics such as those dictatingchemical reactions [139]. On the other hand, few experiments have attempted to control thenon-equilibrium states of systems undergoing mixed dynamics. Indeed, recent theoreticalwork in the field of quantum control has actively investigated questions related to such openquantum systems (e.g. [13, 132, 39]). Formulating optimal control protocols can be chal-lenging for such systems. In fact, it is usually difficult to even decide whether or not an

CHAPTER 5. REALISTIC AND VERIFIABLE COHERENT CONTROL 86

open quantum system is controllable (i.e., whether all states are reachable from an arbitraryinitial state), given a model of its dynamics and control [39]. Theoretical treatment of suchsystems is complicated by the fact that one cannot exploit a clear separation of timescales torestrict the dynamical model to purely coherent or incoherent dynamics. Instead one mustincorporate both coherent dynamics and decoherence processes in a unified manner, e.g.,using quantum master equation models.

In this paper we examine a paradigmatic open quantum system, electronic excited statesin photosynthetic light harvesting complexes (LHCs) [8], and investigate strategies for con-trolling the non-equilibrium dynamics of these excited states. Photosynthetic light harvest-ing complexes are typically composed of dense arrangements of pigment molecules, such aschlorophyll, embedded within protein backbones. Electronic excited states result from theabsorption of photons by pigments in such pigment-protein structures. These states – termedexcitons – can be either localized on single pigments or delocalized across multiple pigmentsdue to the strong electronic coupling between pigments. The excitation energy carried bythese states is funneled to regions of the light harvesting complex that can initiate the decom-position of such excitons into separated free charge carriers. This energy transfer process,which is dictated by the dynamics of excitons, is extremely complex and has recently beenshown to have significant quantum coherent character [44, 37, 108]. Subsequent modelingand theoretical study [96, 111, 75, 117, 26, 125, 121, 33, 27, 34, 25] have determined thatthe energy transfer process is described by a finely tuned balance of coherent and incoherentdynamical processes. Due to this partially coherent nature of the excitonic dynamics, thecontrol of the energy transfer process in LHCs using laser fields is expected to be sensitive toboth the temporal and spatial phase coherence of the controlling laser fields. For these rea-sons, we regard the control of energy transfer in LHCs as a paradigmatic example of coherentcontrol of mixed quantum dynamics in open quantum system dynamics. Achieving control ofexcitonic dynamics in photosynthetic systems is a first step towards active control of energytransfer dynamics in complex organic molecular assemblies, a potentially important capabil-ity for artificial light harvesting technologies [124]. The ability to control excitonic dynamicsin LHCs is also a potentially useful tool in the quest to develop a complete understandingof energy transfer in these complex systems.

Several previous studies have already addressed the coherent control of excitonic dynam-ics in LHCs. Herek et al.performed an early experiment demonstrating moderate controlover energy transfer pathways in the LH2 light harvesting complex using shaped femtosec-ond laser pulses [63]. Theoretical modeling of this same LHC and calculation of optimalcontrol fields to achieve enhanced fluorescence was performed in Ref. [85]. These studiesdemonstrated coherent control of chemical products of light harvesting system, but not con-trol of femtosecond electronic dynamics themselves. In contrast, Brüggemann, May andco-workers performed a series of theoretical studies focusing on the formulation of optimalfemtosecond pulses to control the excitonic dynamics of the Fenna Matthews Olson (FMO)complex [16, 18, 17]. These studies aimed to localize excitation energy in regions of theFMO complex using shaped pulses with and without polarization control. Recently, Carusoet al.[24] performed a theoretical study that focused on preparing various localized and prop-


agating excitonic states of the FMO complex using shaped femtosecond pulses determinedwith the recently introduced CRAB optimization algorithm [40].

In this work, we extend the studies of optimal control of excitonic dynamics in FMO bysystematically analyzing the limitations to achievable control that are imposed by practicalconstraints for bulk, small ensemble and single complex experiments. As an importantcomplement, we also identify schemes for authentication of any such coherent control ofexcitonic dynamics by prediction of the signatures of coherent control in pump-probe spectra.We close with a discussion of the outlook for further development of coherent control andits applications for analysis of excitonic coherence in biological systems.

5.2 Model for laser excitation of FMOWe use the Heitler-London model Hamiltonian given by Eq. (1.1–1.6).

We focus our investigations on the FMO complex of green sulfur bacteria, an extensivelystudied protein-pigment complex. Biologically, a monomer of the FMO complex serves as a“quantum wire” with seven chlorophyll pigments that transfers excitation energy from thechlorosome antenna towards the reaction center in a partially coherent manner [44, 37].The crystal structure of the FMO protein is known, and the system has been subject toa large number of linear and non-linear spectroscopic measurements [94]. Accordingly, theHamiltonian of the system is quite well established. Here we use the electronic Hamiltoniandetermined by Adolphs and Renger that includes Gaussian static (ensemble) disorder of full-width-at-half-maximum 100 cm−1 for each transition energy En [2]. We model the system-reservoir coupling by assuming that each pigment is coupled to an independent reservoirwith identical Debye spectral densities Jn(ω) = J (ω) of the form ω2J (ω) = 2λγω

ω2+γ2Θ(ω),

where Θ denotes the Heaviside function and with reorganization energy λ = 35 cm−1 andbath relaxation rate γ = 1/(50 fs) tuned to experimental values as in Ref. [75]. Because forFMO all of the key energy scales in the problem coincide (~λ ∼ ~γ ∼ Jnm), the Hamiltoniangiven by Eqs. (1.1)-(1.6) does not fall in the range of validity of the standard perturbativeapproaches of either Förster or Redfield theories [75]. This has made photosynthetic sys-tems, and in particular the FMO complex, prototypes for alternative methods to solve openquantum systems [76, 122, 72, 146, 114, 33] These methods provide significant improvementin accuracy, including, for example, representation of non-Markovian effects.

Nonetheless, in this work we use Redfield theory in the secular approximation, becausewe can evaluate Redfield dynamics for a single molecule of FMO in ∼ 100 ms on a singlemodern CPU, compared to up to hours for more accurate methods [75, 122, 72, 146, 114,33]). Such rapid evaluation of dynamics solutions is a necessary trade-off to ensure we canoptimize candidate control pulses in a reasonable time. However, we do not expect ouroptimally shaped pulses would show qualitative differences for controlling exciton dynamicswhen evaluated on more realistic models. The approach of secular Redfield theory is to treatthe system-reservoir coupling HS-R to second order and to apply the Markov and secularapproximations [93]. This allows us to write an equation of motion for the electronic reduced


density matrix ρ,

dρ

dt= − i

~[HS +HS-L(t), ρ] +Dρ (5.1)

where the dissipation superoperator D can be written as a sum of components in the standardLindblad form. We include the explicit time-dependence in HS-L to indicate that this is theonly non-constant term (aside from ρ), which holds under the assumption that we can neglectmodification of the dissipative dynamics due to strong field excitation [128]. This descriptiongives rise to qualitatively reasonable dynamics including quantum beats and relaxation to theproper thermal equilibrium, unlike other efficient simplified quantum master equations suchas the Haken-Strobl or pure dephasing model [55, 23]. Employing the secular approximationguarantees completely positive dynamics, at the price of neglecting environment induceddynamical coherence transfer [107].

To efficiently simulate light-matter interactions, we use the rotating wave approximation,which allows us to ignore the contribution of very quickly oscillating and non-energy conserv-ing terms [140]. To do so, we switch to a rotating frame with frequency ω0, typically chosento match the carrier frequency of the applied field. The dynamics are given by replacing eachHamiltonian H with its rotating equivalent H. The only terms in Eqs. (1.1)-(1.6) which arenot equivalent to those in the non-rotating frame is the system Hamiltonian HS, which in therotating frame has the transition energies En = En − ~ω0, and the system-light interaction

HS-L =∑n

(dn · e) anE(t) + h.c. (5.2)

The electric field in the rotating frame, E(t), is related to the full electric field by E(t) =

eE(t)eiω0t+c.c., where e is a unit vector in the direction of the polarization of the field. Sincethe Redfield dissipation superoperator D is time-independent, to determine the electronicdensity matrix ρ resulting from a control field we may numerically integrate Eq. (5.1) uponsubstituting HS and HS-L for HS and HS-L, respectively. Weak fields also allow us to performefficient averages over all molecular orientations [38].

5.3 Optimal control methodsWe focus here on optimizing state preparation under weak field excitation, the experimen-tally relevant regime for ultrafast spectroscopy. Spectroscopic measurements usually cannotdetermine the fraction of an ensemble sample that is excited, so we employ the normalizedexcited state density matrix given by

ρ′ =ρe

Tr ρe, (5.3)

where ρe denotes the projection of the density matrix onto the single excitation subspace.In the weak field regime, ρ′ will not be sensitive to the overall pump amplitude, since thetotal excited state population will remain small.


Our control objectives can all be expressed in terms of this normalized excited statedensity matrix ρ′. Our first state preparation goal is localization of the excitation on apigment n:

maxCsiten (t) ≡ max〈n|ρ′(t)|n〉 (5.4)

for n ∈ {1, 2, . . . , 7}, with a†nan|m〉 = δmn|m〉. The second goal we consider is the preparationof an excitonic state:

maxCexcα (t) ≡ max〈α|ρ′(t)|α〉 (5.5)

for α ∈ {1, 2, . . . , 7}, where the excitonic states |α〉 are the eigenstates of HS, and we chooseto order them by energy so that exciton α = 1 is the lowest energy one. The third goal isthe maximization of an excitonic coherence:

maxCcohα,β (t) ≡ max |〈α|ρ′(t)|β〉| (5.6)

for α, β ∈ {1, 2, . . . , 7}. The time t in all these cost functions is restricted to times followingthe end of the shaped control pulse. In addition to these state population targets, in section5.5 we describe and carry out optimization of another cost function of ρ′(t) that is directlyrelated to experimentally measurable spectroscopic signatures.

We employ two types of pulse parameterizations in this work, optimizing the param-eterized control fields in each case with standard numerical optimization techniques (seebelow). To first determine the theoretical limits of control without taking any constraints onphysical realization of the control fields, we use an adaptation of the chopped random basis(CRAB) scheme [22], which was recently used to optimize state preparation for FMO [24].This control field is defined in the time domain and is of the form

E(t) = fcutoff(t)N∑k

(Ak + iBk) eiωkt, (5.7)

for N fixed frequencies ωk and 2N real valued optimization parameters Ak and Bk. This pa-rameterization allows variation of both amplitude and phase of the control field components.In our case, we choose N = 19 frequencies ωk at equal intervals spanning the FMO absorp-tion spectrum. The cutoff function fcutoff(t) is chosen as a step function that restricts thecontrol fields to a particular total pulse duration T . For complexes with fixed orientation (incontrast to optimization over an ensemble of orientations), we also include parameters φ andθ to choose the optimal polarization e of the control field. These optimization parametersare illustrated in Figure 5.1.

The parametrization given by Eq. (5.7) creates pulses with different overall energiesand energy distributions, but “coherent control” is in some cases understood as referringto exploiting the coherent nature of the laser light itself [140]. For this reason, some initialexperimental demonstrations of coherent control have stressed the sensitivity of control to


�

✓

Figure 5.1: Parameters used in the first parameterization [Eq. (5.7)] for controlling excitationof FMO, superimposed over the FMO linear absorption spectrum. The vertical red linesindicate the chosen frequencies ωk.

the phase of the control field [115]. Thus, we also consider a second pulse parametrizationsuited for coherent control within an experimentally constrained phase-only control scenario.Here we define the control field in the frequency domain and each frequency is assigned avariable phase term which is parameterized as a polynomial function of the frequency:

E(ω) = A(ω) exp

[iN+1∑k=2

Ck(ω − ω0)k

]. (5.8)

The quantity A(ω) denotes the unshaped amplitude profile of the control field, which isfixed by the laser setup, ω0 is the central frequency of the laser and Ck is a real valuedoptimization parameter. We neglect terms with k < 2, in order to remove global phase shifts(k = 0) as well as time-translations (k = 1). The surviving terms k ≥ 2 correspond to linear(k = 2) and higher-order (k > 2) chirps in the frequency domain. In this work we havechosen the amplitude A(ω) to be represented by a Gaussian function, with amplitude fixedto ∼ 107 Vm−1 in the time-domain and standard deviation 225 cm−1, corresponding to afull-width-at-half-maximum of 55 fs in the time domain. In this paper, we restrict ourselvesto N = 10 terms. To simulate hypothetical pulses matching those produced by an ultrafastpulse shaper [130, 149], we apply this field in the frequency domain to 600 points stretchingbetween ±3 standard deviations, and obtain the time-domain pulse in the rotating framefrom the inverse Fourier transform. Cutting off frequencies more than 3 standard deviationsaway from the central frequency does result in some small artifacts visible in the time-domain. We then use the following two-step heuristic to determine the final time for thepulse: we set to zero the pulse field at all times with amplitude less than 0.005 times themaximum amplitude and then set the final time as the point at which 99% of the overall


integrated total pulse amplitude has passed. For optimization of oriented complexes, therelative polarization e is specified by angles θ and φ as above.

To optimize our pulses over the parameters in Eqs. (5.7) and (5.8) we use two opti-mization routines, the Subplex direct search algorithm [123] and the Covariance MatrixAdaptation Evolution Strategy (CMA-ES) [58] genetic algorithm. Subplex is a local, deriva-tive free search algorithm that has been previously used with control fields using the CRABparametrization [22, 24]. CMA-ES is a genetic algorithm designed for solving difficult con-tinuous optimization problems while employing only a few tunable hyper-parameters (herewe set the population size to 50 and the initial standard deviation to 1). We limited eachoptimization algorithm to a maximum of 10 000 function evaluations with a single initialcondition, Ak = Bk = 1 for Eq. (5.7) and Ck = 0 for Eq. (5.8). Interestingly, we foundthat the best optimization algorithm depended on the particular optimization target (e.g.,Subplex worked better than CMA-ES for 69% of the cases depicted in Fig. 5.3), so we usedthe best result from optimizations with both algorithms. With more function evaluationsavailable, more random initial conditions could be used with Subplex (as in Ref. [24]) andpopulation sizes could increased for CMA-ES. Because both algorithms struggle to optimizeover very rough control landscapes, when implementing using the pulse parameterization inEq. (5.7), we choose the total pulse duration with a separate optimization, by performinga grid search over all pulse durations between 100 fs and 500 fs at increments of 50 fs. Inpractice, we find this significantly increases the consistency and quality of our optimizationresults compared to including the pulse duration as a search parameter. In addition, toguarantee that our optimal pulses are still in the weak field regime with Eq. (5.7), we sub-tract from each objective function a smooth cutoff function of the form Θ(x − α)(x − α)2,where x is the total excited state population and α = 0.01 is set as the maximum acceptableexcited state population. Note that for an unshaped Gaussian pulse (i.e., Eq. (5.8) with allCk = 0), this maximum excited state population of 1% corresponds to a maximum electricfield amplitude in the time domain of no more than 1.2× 107 Vm−1.

To match experimental conditions, in addition to optimizing over pulse parameters, inmost of our optimizations we average over ensembles of complexes characterized by orienta-tional and/or energetic (static) disorder. For these calculations, we use two approaches. Toaverage over the orientations, we use the fact that the isotropic average of the excited statedensity matrix under weak fields is equal to the average of the x, y and z polarization ori-entations (see Appendix 5.7.1). Accordingly, we can obtain the exact orientational averageat the cost of only a factor of 3 times more function evaluations. Unfortunately, there is nosuch shortcut for averages over energetic disorder. In such cases, we perform optimizationson the average of 10 randomly (but consistently) chosen samples, and still limit ourselvesto the fixed computational cost of 10 000 function evaluations by now allowing for no morethan 1000 function evaluations. To determine final optimal results in the presence of staticenergetic disorder, we average over ensembles of 1000 samples. For these simulations, we av-erage the excited state density matrix ρe in the site basis before inserting it into the objectivefunctions given by Eqs. (5.3–5.6).


5.4 Limits of coherent control for initial statepreparation in FMO

An interesting question to consider before investigating control pulses designed for specifictarget states, is whether or not the FMO system can be prepared in any arbitrary state, i.e.,whether FMO is controllable. Formally, a closed quantum system is completely controllableif control fields can be used to generate any arbitrary unitary operation [127]. Complete con-trollability implies that it is possible to prepare arbitrary pure states starting from any initialpure state, which in our case is the ground state. The controllability of a closed quantumsystem can be analyzed algebraically, by examining the Lie algebra generated by the systemand control Hamiltonians. Accordingly, we consider the following highly idealized scenario:the electronic degree of freedom for a single FMO molecule without coupling to vibrationsand restricted to at most one electronic excitation. Inter-pigment and pigment-protein in-teractions break the degeneracy of the excited states, so for the system without dissipationthere actually is a constructive algorithm for determining arbitrary unitary controls [116].We confirmed this by running the algorithm of Ref. [127] to verify that we do indeed havecomplete controllability for formation of exciton states in the FMO complex under the com-bination of the electronic system Hamiltonian HS and the light-matter interaction HS−Lwith any arbitrarily chosen single polarization of light.

In contrast, there are few rigorous results known for determining the controllability ofopen quantum systems [39]. Accordingly, in our calculations below for realistic experimentalconditions (weak fields, bulk samples with orientational and static disorder, finite tempera-tures) we assess the feasibility landscape of both arbitrary (amplitude/phase) and phase-onlycontrol in a brute-force manner, namely by evaluating the effectiveness of candidate controlpulses for various targets. Our constraints are chosen to reflect the current experimentalsituation and include the use of weak fields, averaging over all orientations, averaging overstatic disorder and decoherence at 77K. Averaging over orientations and over static disor-der would not be necessary in single complex spectroscopy experiments, which may becomefeasible in the forseeable future (see below). We have carried out optimizations to bothmaximize and to minimize population on specific sites and excitonic states.

The results of these optimizations with amplitude and phase control [Eq. (5.7)] and withphase-only control [Eq. (5.8)], are summarized in Figure 5.2 and compared with the corre-sponding populations achieved using unshaped pulses. The latter is the set of populationsrealized by a fixed “unshaped” Gaussian pulse with a variable time-delay after the end ofthe pulse. The variable time-delay simply allows incorporation of the inherent relaxation inthe system, which can aid in optimizing some goals (e.g., preparation of the lowest energyexciton, or of site 3, on which the lowest exciton is mostly localized). These results show thatunder experimental conditions corresponding to bulk samples of FMO complexes in solutionat liquid nitrogen temperatures, we are in a coherent control situation where fidelity valuesare more similar to those achieved in typical reactive chemical control situations than thevalues required for quantum information processing, underscoring once again the critical role


1 2 3 4 5 6 7Excitons

0.0

0.2

0.4

0.6

0.8

1.0

Popu

latio

n ra

nge

Phase & amp.Phase onlyUnshaped

1 2 3 4 5 6 7Sites

Figure 5.2: Population control results for all targets (site and exciton populations). Blueand red bars indicate the minimum-to-maximum populations achievable with phase onlycontrol and with phase plus amplitude control, respectively. The white bars indicate theminimum-to-maximum achievable by using unshaped pulses together with a variable timedelay following the pulse (see text). Phase only control is a special case of phase andamplitude control, and unshaped pulses are a special case of shaped pulses, so the red areais actually inclusive of the entire range shown and the blue area is inclusive of the white bar.Exciton 1 has the lowest energy and exciton 7 the highest energy.

of the open quantum system dynamics.We now consider the effect of each of three major constraints on the achievable fidelity of

control, both singly and in all possible combinations. We identify these three primary con-straints as the following: the isotropic average over all molecular orientations, the ensembleaverage over static disorder of the transition energies En and the effects of decoherence due tothe interaction with the environment (represented by a bath of phonon modes) at 77K. Theindependent and cumulative effect of these constraints are illustrated by the Venn diagramsin Figure 5.3 and the values are given in Table 5.1. We have restricted the depictions hereto just the results for maximizing each exciton or site population and for joint phase andamplitude control using Eq. (5.7).

Although it is clear from the lower panels of Figure 5.3 that the precise results of opti-mization are different for each target state, we can nevertheless extract a number of strikingtrends. For example, for most targets, the single largest limiting factor to controllability isdecoherence. Also, in general, it is easier to achieve single exciton rather than site localizedtarget states, as indicated by the average control fidelities over all sites or all excitons inTable 5.1. Site 3 is an exception to both these trends, which is readily rationalized by recog-nizing that the lowest energy exciton (exciton 1) is primarily localized on site 3. Indeed, it isnotable that optimization both of population on site 3 and in exciton 1 are relatively robustto all three of the constraints. We can also see that averaging over orientations without


Site 1:

Isotropicaverage

Ensembledisorder

Decoherence(at 77 K)

Closed system,oriented,

single molecule100%

84.8% 72.4%

63.9%

57.4%

50.7%44.9%

32.9%

0.2

0.4

0.6

0.8

1.0

Tar

getf

idel

ity

1 2 3 4 5 6 7

Site

Exc

iton

Figure 5.3: Venn diagrams indicating control fidelity for optimization under different exper-imental constraints with joint phase and amplitude control. The outer region refers to theclosed quantum system constituted by the excitonic Hamiltonian without coupling to theprotein environment. Here the excitonic system is completely controllable and the fidelityof preparing any state is 100%. The circles enclosed by colored lines indicate the fidelityachieved when we add averaging over orientation of the FMO complex (red circles), averagingover site energy disorder (green circles) and adding the coupling to the protein environmentat a finite temperature (blue circles). (Above) Venn diagram for maximum population onsite 1. (Below) Venn diagrams for all site and exciton targets.


Unitary Decoherence (at 77 K)Single Ensemble Single Ensemble

Target Orient. Iso. Orient. Iso. Orient. Iso. Orient. Iso.

Site

1 100.0 84.8 72.4 57.4 63.9 44.9 50.7 32.92 100.0 91.1 86.4 65.2 74.3 37.6 58.7 30.63 100.0 97.2 94.0 79.1 87.8 87.9 74.9 69.44 100.0 83.9 71.8 48.2 57.5 48.7 47.3 35.25 100.0 88.0 74.5 55.0 71.2 40.7 51.7 35.26 100.0 92.4 94.9 81.6 87.3 59.6 80.1 45.77 100.0 92.6 84.7 31.3 74.5 23.8 62.2 16.3

Avg. 100.0 90.0 82.7 59.7 73.8 49.0 60.8 37.9

Exciton

1 100.0 100.0 96.6 95.3 92.4 89.9 86.5 84.22 100.0 100.0 89.4 83.6 76.1 59.0 68.6 52.53 100.0 100.0 89.3 66.7 79.4 55.8 62.4 44.04 100.0 99.9 92.2 41.9 81.5 32.7 75.3 24.45 100.0 99.9 86.9 68.1 73.1 48.3 61.5 39.06 100.0 100.0 92.7 87.2 64.9 49.1 53.5 34.17 100.0 100.0 98.4 97.2 90.5 76.0 84.9 51.1

Avg. 100.0 100.0 92.2 77.1 79.7 58.7 70.4 47.1

Table 5.1: Control fidelities for the optimizations illustrated in Fig. 5.3.

decoherence is evidently a more severe constraint for localized site target states than forsingle exciton target states. This can be rationalized as reflecting a simple physical strategyfor targeting excitonic states, namely, to drive the system for a long time at the proper tran-sition energy. In most cases, it is clear that the difficulty of control under realistic conditionsstems from the combination of two or more of these limiting factors, which together rule outmany intuitive control strategies.

The optimal pulse shapes for selected instances of site and exciton target states, withand without ensemble averaging over static disorder, are shown in Figure 5.4. We illustratethese pulses by plotting their Wigner spectrograms [99], given by the expression

W (t, ω) =

∫ ∞−∞

dτE∗(t− τ/2)E(t+ τ/2) exp(iωτ), (5.9)

where E(t) is the pump field and ∗ denotes the complex-conjugate. The Wigner spectro-gram can be interpreted as a type of joint time-frequency distribution, with the desirableproperties,

|E(t)|2 =

∫dω

2πW (t, ω) (5.10)

|E(ω)|2 =

∫dtW (t, ω). (5.11)


1.15

1.20

1.25

1.30

1.35

Freq

uenc

y (1

04 c

m−

1)

Exciton 6, single Exciton 6, ensemble

0 100 200 300 400 500Time (fs)

1.15

1.20

1.25

1.30

1.35

Freq

uenc

y (1

04 c

m−

1)

Site 1, single

0 100 200 300 400 500Time (fs)

Site 1, ensemble

Figure 5.4: Optimal pulses to maximize the population of the isotropic average on exciton6 (top) or site 1 (bottom) in the presence of decoherence, both without (left) and with(bottom) static disorder. The central plot of each pulse is of the Wigner spectrogram. Red(blue) indicates positive (negative) values. Top and right plots show the marginal time andfrequency distributions |E(t)|2 and |E(ω)|2.


0 20 40 60 80 100Reorganization energy (cm−1 )

0.0

0.2

0.4

0.6

0.8

1.0

Targ

et p

opul

atio

n (s

ite 1

)

0 50 100 150 200 250 300Temperature (K)

Single moleculeEnsemble disorder

Figure 5.5: Maximum target population (site 1) as a function of the bath reorganizationenergy (left) and temperature (right), hence modifying the amount of decoherence in theexcitation dynamics. The results are shown in the orientational average with and withoutincluding ensemble disorder. The standard bath parameter choices, used when not modifyingthe decoherence rate, are indicated with the vertical dashed lines. Left: Temperature is fixedat 77K. Right: Reorganization energy is fixed at 35 cm−1.

Recall that because we are in the the weak-field regime and normalize all objective functionsby the excited state population, our results are independent of the overall amplitude of thepump pulse, so there is no need to provide an absolute scale for the control field amplitude.As Figure 5.4 shows, optimal pulses for targeting states in the presence of static disordergenerally require a more complicated frequency distribution and a broader band of excitationfrequencies. This is particularly evident from the optimal pulses for creating an excitationin a single excitonic state, where the optimal pulses are generally peaked at the transitionenergy of the desired state. The optimal pulses for preparing an excitation in exciton 6 (toppanels) illustrates also the complementary trend that in the presence of static disorder theoptimal pulses are usually shorter in time. Not surprisingly, pulses designed with optimalpolarizations (i.e., without the average over all orientations), are even shorter since theseoptimizations can make use of polarization in addition to or instead of excitation frequencyto selectively target specific states.

To explore the influence of decoherence in more detail, we consider the independent effectsof adjusting the temperature of the bath and the reorganization energy. Figure 5.5 showsthe error for maximizing population in one specific site, as a function of the reorganizationenergy (left panel) and as a function of the bath temperature (right panel). We see thatalthough lowering the bath temperature to near zero does make it easier to target site 1,this is not as powerful a control knob as removing the system-bath coupling (i.e., setting thebath reorganization energy to zero).


5.5 Authentication of coherent control in pump-probespectra

To experimentally verify the excited state that is created by a control pulse, we propose touse ultrafast pump-probe spectroscopy. In pump-probe spectroscopy, a pump pulse excitesthe system, followed by a probe pulse at a controlled delay time. Although there are anumber of ultrafast spectroscopies that may be used for probing light harvesting systems[98], pump-probe spectroscopy is particularly well suited for verifying state preparation sinceits signal can be formally interpreted as a function of the full excited state created by thepump pulse [69]. In the limit of an impulsive probe pulse, provided that we are in the weakfield regime and that there is no time overlap between the pump and probe pulses, thefrequency resolved pump-probe signal is given by

S(ω, T ) ∝ Re{

Tr[µG(ω)[µ†, ρ(2)pu (T )]]

}, (5.12)

where ω is the probe frequency and T is the time delay of the probe relative the end of thepump probe [69]. We emphasize that one cannot verify state preparation with this approachbefore the end of the pump pulse. The operator G(ω) is the Fourier transform of the retardedmaterial Green function, µ is the annihilation component of the dipole operator µ in thedirection of the probe field and ρ(2)

pu (T ) is the second order contribution to the density matrixin terms of the pump pulse, restricted to the 0-1 excitation manifold. For a Markovian bathand weak excitation, this density matrix ρ(2)

pu (T ) can be written as a linear function of theexcited state density matrix ρe(T ) [69]. Accordingly, we can write Eq. (5.12) in the form

S(ω, T ) ∝ Re {Tr[P(ω)ρe(T )]} (5.13)

where P(ω) is a (linear) superoperator in Liouville space. This equation still holds even fora randomly oriented ensemble for a pump-probe experiment performed in the magic-angleconfiguration [65], provided that P(ω) and ρe are replaced by their isotropic averages (seeAppendix 5.7.1). We then efficiently perform each of these independent isotropic averagesby averaging over the x, y and z pump (or probe) polarizations [38].

The ideal authentication procedure for any quantum control experiment is quantum statetomography, in which each element in the density matrix is determined by repeated mea-surements on an ensemble of identically prepared systems [103]. It is possible to constructprotocols for state tomography of excitonic systems that are based on pump-probe spec-troscopy [157, 69], but the uncertainty concerning Hamiltonian parameters, together withthe averaging over energetic disorder and orientations associated with ensemble measure-ments, make this an unrealistic target for systems with as many pigments as FMO [69]Accordingly, here we consider two options for authenticating control that may be realizedwith current experimental technology. The first is to extend the duration of coherent elec-tronic beating and the second is to generate a novel signature in the pump-probe signal. Wenote that with a frequency resolved signal as given by Eq (5.12) there are no gains from


optimizing the probe pulse, as long as it is faster than the dynamics of interest and hasnon-zero amplitude at the frequencies of interest.

Although the evidence for electronic coherence today is more clearly evident in 2D photon-echo spectroscopy [44, 37], the first evidence for quantum beats in excitonic dynamics ofthe FMO complex were actually found with pump-probe spectroscopy [126]. Under secularRedfield dynamics, the excitonic coherence terms are decoupled from all other dynamicsand decay exponentially. Hence to maximize the duration of quantum beating within thisdescription of the dynamics, we merely need to maximize the absolute value of the coherenceterms in the excitonic basis. However, predicting the visibility of electronic quantum beats iscomplicated by the relative orientations of the relevant transition dipole moments, as well asby the need to observe any oscillations before they decay. Accordingly, we demonstrate herethe result of numerically maximizing the excitonic coherence |ρ12| between the lowest twoexciton energies, since observations of quantum beats in pump-probe spectra have previouslybeen ascribed to this coherence [126]. (We do not optimize for the ensemble value of thiscoherence, because the ensemble average of excitonic coherences in the eigenbasis of theHamiltonian without static disorder do not necessarily decay to zero at long delay times.) Forthis optimization we use the unrestricted parametrization of Eq. (5.7), in order to determinethe limits of maximizing this signal. We note that the aforementioned experimental pump-probe study did use a form of incoherent control to maximize the this signal, by scanningthe central frequency of the pump pulse [126]. Our results show that we can increase themagnitude of the excitonic coherence |ρ12| by a factor of up to 8.6x compared to an unshapedGaussian pump. Figure 5.6 illustrates the optimal pulse and its signature on the pump-probe spectra including ensemble disorder. As shown in panel (a), the increased coherenceresults in increased oscillatory features that will be visible in a magic angle pump-probeexperiment, particularly after subtracting away the non-oscillating contribution to the signal[panel (b)]. This subtraction of the non-oscillating signal is similar to the technique used in[108] to estimate the timescale of quantum beats in a 2D photon-echo experiment. Even afterincluding static disorder, the beating signal following the optimized pump shows oscillationswith a magnitude about 3x larger than those following the original, unshaped pump pulse.The optimal pulse (c) uses the unsurprising strategy of sending a very short pulse peaked atthe relevant exciton transition energies, ω1 = 12 170 cm−1, ω2 = 12 282 cm−1. Although ouroptimal pulse uses a total duration of T = 350 fs, all of its meaningful features are in the last100 fs; indeed, our optimizations for different total times between 100 fs and 500 fs showedonly a small variation in the factor of enhancement of the 1-2 coherence, ranging from 8.4to 8.6.

To further demonstrate the feasibility and authentication of coherent control of energytransfer in FMO, we now consider using phase-only control to create new states that are notaccessible from unshaped (Gaussian) pulses. This optimization for new states with phase-only control proved to be quite difficult. In most cases, phase-only control only enhances siteor exciton populations by at most a few percent in comparison to the unshaped pulse (seeFigure 5.2). Verifying such small differences in the population of a specific site or exciton isalmost certainly beyond the current experimental state of the art for these open quantum


0 200 400 600 800 1000Delay time (fs)

8

7

6

5

4

3

2(a)

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.u.)

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0 200 400 600 800 1000Delay time (fs)

0.2

0.1

0.0

0.1

0.2 (b)

Bea

ting

sign

al

Unshaped

Optimized

0 100 200 300 400 500Time (fs)

1.15

1.20

1.25

1.30

1.35(c)

Freq

uenc

y (1

04cm

−1

)

Figure 5.6: Enhanced coherent beating in the pump-probe signal. (a) Differential absorption[the pump-probe signal given by Eq. (5.12)] at a probe frequency of 12 200 cm−1 followingunshaped (solid blue line) or optimized pump (dashed red line) pulses, normalized by thetotal excited state population following the pump pulse. The probe frequency was chosen asan arbitrary value, in the range of the exciton 1 and 2 transition energies where the oscillatorysignal from the optimized pump is clear. Dots show a double exponential fit for each signalof the form Ae−at + Be−bt + C, where A, B, C, a and b are constants chosen to minimizethe least squares distance between the fit and the differential absorption signal. (b) Beatingportion of the signal shown in panel (a) after subtracting the double exponential fits, A =3.13 cm−1, B = 1.79 cm−1, C = −6.89 cm−1, a = 3.11 ps−1, b = 1.29 ps−1 for the unshapedpulse and A = 44.8 cm−1, B = −45.4 cm−1, C = −6.76 cm−1, a = 5.57 ps−1, b = 5.52 ps−1 forthe optimized pulse. (c) Wigner spectrogram showing the time-frequency distribution of theoptimized control pulse, depicted in the same style as the pulses in Fig. 5.4.


systems [69]. Achieving phase-only control is especially difficult for natural light harvestingsystems because they feature rapid dephasing and energetic relaxation, with each of theseoperating at a similar time scale to the natural time scale of the energy transfer (see above).Applying phase-shaping by necessity increases the temporal duration of the pulse, and thusin most cases simply drives the system closer to the equilibrium excited state, rather thanto a selected state.

The exact dependence of the pump-probe signal on the density matrix ρ(2)pu (T ) given by

Eq. (5.12) is difficult to determine with high accuracy, even with the assistance of exper-imental input. Nevertheless, the formal dependence of the signal on the density matrix[Eq. (5.13)] means that this can be used as a witness for coherently controlled formationof new target states. Consider the reference set of states through which the system passesfollowing the end of an unshaped Gaussian pump pulse. Our goal is now to create a statewith a spectrum [Eq. (5.13) with any fixed value of T ] that falls outside this reference set.We achieve this by directly optimizing the difference in a simulated pump-probe spectrumfrom all possible spectra obtained from the reference set of states. With this procedurewe found that it is possible to create a 12.0% relative difference in the signal outside therange of spectra resulting from an unshaped pulse with a time delay. The resulting spectraand visualizations of the optimal pulse constructed from Eq. (5.8) with N = 10 terms areshown in Figure 5.7. We chose N = 10 because our optimizations seemed to reach a point ofdiminishing returns; in fact, we were able to achieve a maximum difference of 4.2% with first-order chirp alone (N = 1) and 9.8% by adding only one higher order chirp term (N = 2).Inspection of the optimal pulse [panels (b-d)] reveals that this is acting to maximize thepopulation in low energy states (excitons 2 and 3) by sending in the higher energy light first,so that the excitons absorbed at these frequencies have time to relax to lower energy states.The population in these states is then further amplified by absorption of a final burst of lowenergy light just before the pulse ends.

5.6 Discussion and outlookIn this paper we have considered the simplest experimental setup for coherent control ofexcited state dynamics in light harvesting systems, namely a pump-probe configuration.Most importantly, we showed that in a realistic experimental scenario it should be possibleto do two types of coherent control and verification experiments: to create of provably newstates with phase-only control, and, if both phase and amplitude shaping are available, toenhance quantum beatings.

We also analyzed in detail the effect of various restrictions to understand what limitscontrol in light harvesting systems. The reference closed quantum system given by justthe seven chromophores of a single FMO complex is completely controllable, but when thissystem is treated correctly as an ensemble of open quantum systems, the extent of control ofthe excitonic subsystem is significantly reduced. Our systematic investigation of the limitsof coherent control under the constraints of decoherence at finite temperatures, orientational


25

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10

5

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10

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0

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10(b)

Pum

p fie

ld (a

rb. u

nits

)

1.20 1.22 1.24 1.26 1.28012345678

(c)

Frequency (104 cm−1 )

Pum

p fie

ld (a

.u. /

×2π

)

Amplitude

Phase

0.0 0.2 0.4 0.6 0.8 1.0 1.2Time (ps)

1.201.211.221.231.241.251.261.271.28

(d)

Freq

uenc

y (1

04cm

−1

)

Figure 5.7: Signature of a novel state in the isotropic pump-probe spectra achieved byphase-only coherent control. The gray area in panel (a) indicates the range of pump-probespectra (black lines) at various time delays (0, 75, 150, 300, 600 and 1200 fs, and in thelimit T →∞) after the end of the unshaped pump pulse, and the red dashed line shows thespectrum immediately following the optimal pulse. The time-domain, frequency-domain andWigner spectrogram of the control pulse are shown in panels (b), (c) and (d), respectively.The dashed line in panels (b) and (d) indicates the temporal end of the control pulse. Thescale on each plot is in arbitrary units, except for panel (c) which plots the phase in multiplesof 2π.


disorder and static (energetic) disorder, revealed that together these three factors restrictthe fidelity of achieving both excitonic (energy eigenstates of the electronic Hamiltonian)and site localized target states. Interestingly, our results show that significant improvementsin extent of controllability should be possible with experiments on well-characterized singlemolecules. Such single-molecule experiments based on non-linear fluorescence may indeedbe possible in the near future [89, 64].

We expect that optimal control will be an even more powerful technique when appliedto more complicated non-linear spectroscopy experiments. For example, simultaneous op-timization of the first two pulses in a third order photon-echo experiment would allow forpreparing particular targeted phase-matched components of the second order density ma-trix ρ(2). Although they ultimately reflect the same third order response function, coherentcontrol of the photon echo could be an attractive alternative to full two-dimensional spec-troscopy. A novel maximally efficient computational scheme for simulating such signals ispresented in Appendix 5.7.2. For higher order spectroscopies the advantages of open loopcontrol for optimal experiment design should be even larger, given that the exponentialgrowth of the state space of possible experimental configurations makes scanning all possibleconfigurations (as in many 2D experiments) increasingly infeasible.

5.7 Appendices

5.7.1 Orientational average

For an ensemble of identical, randomly oriented molecules, it is possible to analytically inte-grate over all possible molecular orientations with a fixed number of system-field interactions.For two interactions (the case for linear spectroscopies), all signals can be written in the form〈S〉 =

∑pq TpqSpq, where each sum is over all orthogonal orientations x, y and z and Tpq is

the second order tensor invariant δpq. For four interactions (the case for all 3rd order spectro-scopies, including pump-probe), all signals can be written in the form 〈S〉 =

∑pqrs TpqrsSpqrs,

where each sum is over all orthogonal orientations x, y and z and Tpqrs is a linear combi-nation of the fourth order tensor invariants, δpqδrs, δprδqs and δpsδqr. The polarization ofthe pump and probe pulses determines which of these tensor invariants contribute to theobserved signal [65]. In the Magic Angle configuration, for which the polarization of the firsttwo interactions is at an angle of 54.7◦ relative to the polarization of the last two interac-tions, only the invariant δpqδrs contributes to the observed signal. Since this invariant is theproduct of the 2nd order tensor invariant for the first and last two interactions separately,the isotropic averages can be performed separately for the first two (pump) and last two(probe) interactions.


5.7.2 Optimal equation-of-motion phase-matching approach forthird-order ultrafast spectroscopy

The formalism for pump-probe spectroscopy that we developed in Sec. 4.2 can be extendedinto an efficient numerical method for simulating arbitrary third-order spectroscopy experi-ments. In the case of the non-rephasing or rephasing geometries for a third-order experimentunder the rotating wave approximation (RWA), this new formalism allows for the most ef-ficient possible calculation of the nonlinear signal field. The technique is a variation of theequation-of-motion phase-matched approach [51, 43, 50] and thus provides a solution that isexact to third order in the strength of the applied fields, unlike some approaches built on thesolution of equations of motion which do not require any such perturbative approximations[91]. We expect this technique could be especially useful for the design of future coherentcontrol experiments using non-linear spectroscopy outside of the pump-probe configuration,as discussed in Sec. 5.6, which is why we include it as an appendix to this chapter. Anotherappealing use would be simulating transient-grating experiments with selective excitationfrequencies, as proposed for a quantum process tomography scheme [157].

To set the context, there are two main types of simulation methods for ultrafast non-linearspectroscopy. In the response function approach [98, 1], the non-linear polarization resultsfrom integrating a non-linear response function over the envelopes of each control pulse.The response function dictates how the system would react to instantaneous (delta-function)control fields. In contrast, equation of motion based approaches determine the non-linearpolarization by performing a number of simulations explicitly including Hamiltonian termsfrom the system-field interaction [51, 50, 91]. Which of these approaches is most suitabledepends on the particular circumstances. For example, the response function approach isparticularly well suited to simulations to simulating 2D spectroscopy, because in its ideal case,2D spectroscopy is performed with broadband pulses so the third-order response functionshould closely approximate the 2D signal. Equation of motion based approaches are moresuitable when the expense of simulating the full response function is unnecessary, such asin the context of simulating a specific experimental setup with non-impulsive control fields.The formulation oft he equation-of-motion phase-matching approach is especially convenientbecause it does not require any assumptions about the time-ordering of the system-fieldinteractions. However, it turns out that knowledge of the time-ordering in some cases mayreduce the computational effort required by a factor of 3.

Our new method involves calculating a density matrix like operator including the firsttwo system-field interactions using a phase-matched equation of motion and then accountingfor the last interaction with either another equation of motion or a linear response function.From Eqs. (1.12–1.13), it follows that in general we can write the experimental signal for athird order experiment under the RWA as

P(3)ks

(t) =

∫ ∞0

dt3R(t3, ρ(2)ks

(t− t3))Eu33 (t− t3), (5.14)


in terms of a probe response function

R(t, ρ(2)ks

) =i

~Tr[µ(−)G(t)V (u3)ρ

(2)ks

], (5.15)

where ρ(2)ks

is given by

ρ(2)ks

(t) =

(i

~

)2 ∞x

0

dt2dt1G(t2)V (u2)G(t1)V (u1)ρ0Eu22 (t− t2)Eu1

1 (t− t2 − t1), (5.16)

in terms of the quantities defined in Sec. 1.2. We note that these equations holds in generaleven without the RWA if µ(±) and V (±) are each replaced by µ and V , respectively.

To calculate the phase-matched contribution to the second order density matrix ρ(2)ks,

we use of equation of motion phase-matching approach of Domcke and co-workers [50].Closely following their argument [50], we note that the phase-matched signal depends onlyon the portion σ2(t) of the system density matrix whose evolution depends on the first twocorrectly phase-matched components. The evolution of σ2(t), initially set to the ground stateρ0, follows the usual equation of motion for the density matrix (any of those described inSec. 1.2) under the time-dependent non-Hermitian Hamiltonian

H1,2(t) = H + v(u1)1 (t) + v

(u2)2 (t) (5.17)

where H is the system Hamiltonian (including any bath or vibrational modes) and

v±a = V (±)λaE±a (t) (5.18)

is an operator for the system-field interaction, where V (±) (V without the RWA) is therelevant transition dipole moment operator, λa is the intensity of pulse a and Ea(t) = E−a (t)is its complex valued electric field [E+

a (t) ≡ E∗a(t)]. Note thatH1,2(t) is non-Hermitian exceptin the case of pump-probe spectroscopy where E1(t) = E2(t) and thus σ2(t) is in general nota valid density matrix. Next, we expand σ2 in a formal power series in terms of the strengthof the first two electric fields λ1 and λ2,

σ2(λ1, λ2) =∑i,j

λi1λj2σ

ij2 (5.19)

=σ002 + λ1σ

102 + λ2σ

012 + λ2

1σ202 + λ2

2σ022 + λ1λ2σ

112 +O(λ3), (5.20)

where for conciseness the time-dependence of each density matrix was removed. Noting thatρ

(2)ks

= λ1λ2σ112 , because that is the second order contribution which depends on both λ1 and

λ2 to lowest order, from Eq. (5.20) we have

ρ(2)ks

= σ2(λ1, λ2)− σ2(λ1, 0)− σ2(0, λ2) + σ2(0, 0). (5.21)

At this point, we see that the second order density matrix can be calculated from at mostthree evaluations of equations of motion under time-dependent Hamiltonians like Eq. (5.17).


Note that we need not perform any integrations to determine σ2(0, 0) = ρ0, since the stateof the system without any applied fields is stationary.

An additional simplification is possible under the RWA, for experiments utilizing thephoton echo or non-rephasing pathways (i.e., excluding the double quantum coherencepathway). For photon echo and non-rephasing pathways, we have u1 = ∓, u2 = ± andu3 = +. Under the RWA, only contributions to ρ(2)

ksthat include equal numbers of rais-

ing and lowering operators can result in diagonal terms that contribute to the trace inEq. (5.15) after applying an additional lowering and raising operator. Accordingly, we haveR(t, σ2(λ1, 0)) = R(t, σ2(0, λ2)) = R(t, ρ0), so Eq. (5.21) may be replaced by

ρ(2)ks

= σ2(λ1, λ2)− ρ0. (5.22)

Thus to calculate ρ(2)ks

under the RWA, we need only integrate the equation of motion given byEq. (5.17) directly. This fact still holds even in a pump-probe experiments where k1 = −k2,since we need only multiply the right hand side of Eq. (5.22) by an additional factor of 2 toaccount for both possible time-orderings of the first two pulses [69].

Once ρ(2)ks

is obtained, the full nonlinear polarization P(3)ks

may be obtained in either oftwo ways. For an impulsive probe pulse, it is convenient to obtain the nonlinear polarizationplugging ρ

(2)kS

from Eq. (5.21) or Eq. (5.22) directly into the response function given byEq. (5.15). The signal under an impulsive probe may be directly useful, because, as notedin Chapters 4 and 5, there appears to be no theoretical advantage to using a non-broadbandprobe pulse. This is the same technique for which we provided an alternative proof for therestricted case of pump-probe experiments in Chapter 4, although that proof required thatthe first two pulses be entirely identical, including their polarization, which restricted thetechnique to the magic angle setup.

Alternatively, for a non-impulsive probe, we construct another time-dependent non-Hermitian Hamiltonian to account for the phase-matched contribution from the last in-teraction,

H3(t) = H + v(u3)3 (t). (5.23)

Because we have a definite time-ordering between the second and third interaction, we canpick an intermediate time t0 when there are no applied fields (i.e., H1,2(t0) = H3(t0) = H)at which to begin applying Eq. (5.23). We then solve for the dynamics σ3(t) under thisHamiltonian with the initial condition σ3(t0) = ρ

(2)ks

(t0). Analogously to the reasoning abovefor σ2, we see that by expanding the state σ3 as a formal power series in terms of the strengthof the last interaction λ3, to lowest order we obtain

ρ(3)ks

= σ3(λ3)− σ3(0). (5.24)

Finally, the third-order non-linear polarization is given by inserting ρ(3)ks

into

P(3)ks

(t) =i

~Tr[µ(−)ρ3

ks(t)], (5.25)


which is the original equation for the non-linear polarization from which one may deriveEq. (5.14–5.16). Under the RWA, Tr[µ(−)σ3(0)] = 0, so we may safely neglect the contributionof σ3(0) to Eq. (5.24) and only integrate under Eq. (5.23) once.

In total, this new approach requires only 1 full numerical integration of an equationof motion (or 3 without the RWA), unlike the 3 (or 7) integrations required when usingthe full equation of motion phase-matched approach [50]. In addition, as noted in Sec. 1.4,restricting dynamics to a particular subset of Liouville space can also be an important formaximum computational efficiency. Under the RWA, we obtain all contributions to the signalby simulating σ2 restricted to the electronic subspaces |g〉〈g|, |g〉〈e|, |e〉〈g| and |e〉〈e|, andσ3 restricted to |g〉〈g|, |e〉〈g|, |e〉〈e| and |f〉〈e|. Since it impossible to numerically simulatedynamics without at least one complete time propagation, our new approach provides forthe most efficient possible calculation of individual pump-probe, photon-echo, non-rephasingor transient grating signals under the RWA, with or without an impulsive probe.

108

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understanding and manipulating electronic quantum coherence in photosynthetic light-harvesting

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